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ALGEBRAIC GEOMETRY in

East Hsia

This page intentionally left blank

Proceedings of the symposium on

ALGEBRAIC GEOMETRY IN

East Hsia Editors

Akira Ohbuchi Tokushima University, Japan

Kazuhiro Konno Sampei Usui Osaka University, Japan

Atsushi Moriwaki Kyoto University, Japan

Noboru Nakayama RIMS, Japan

1,

World Scientific New Jersey London Singapore Hong Kong

Published by

World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: Suite 202,1060 Main Street, River Edge, NJ 07661 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

ALGEBRAIC GEOMETRY IN EAST ASIA Proceedings of the Symposium on Algebraic Geometry in East Asia Copyright 0 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts there05 may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-265-8

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface This book is the proceedings of the conference ”Algebraic Geometry in East Asia” which was held in International Institute for Advanced Studies (IIAS) (9-3 Kizugawadai, Kizu-cho, Soraku-gun, Kyoto 619-0225, Japan), during August 3 to August 10,2001. The conference was partially supported by Grantsin-Aid for Scientific Researches ((A) (1) 11304001 by Sampei Usui and (B) (2) 13440008 by Atsushi Moriwaki). Although many east Asian mathematicians now play a leading role in the international mathematical community, in modern times mathematics did not become a strength of east Asian scholarship until the early 20’th century. In particular, as a consequence of this relatively short history, the various east Asian mathematical communities, such as the algebraic geometers represented at this conference, have had less of a chance to meet and exchange ideas as their western counterparts. Accordingly, one of the primary goals of the conference was to facilitate such an exchange. As the breadth of the topics covered in this proceedings demonstrate, the conference was indeed successful in assembling a wide spectrum of east Asian mathematicians, and gave them a welcome chance to discuss current state of algebraic geometry. It is the first time that such a conference has been held in algebraic geometry, and we hope that it is but the start of continuing tradition. We wish to thank, first of all, the lecturers for their beautiful talks. We also wish to thank the participants for their cooperation and providing stimulating atmosphere. During the conference, administrative staffs in IIAS as well as many graduate students from Osaka University and Kyoto University helped us. Without them, the conference would have been less successful than we had hoped for. In particular, we wish to thank Ms. Yoshiko Kusaki and Ms. Minako Tanaka of the IIAS as well as the following graduate students from Osaka and Kyoto Universities: Mr. Masao Aoki, Mr. Takeshi Harui, Mr. Atsushi Ikeda, Mr. Michiaki Inaba, Mr. Tomokazu Kawahara, Mr. Hiraku Kawanoe, Mr. Shinya Kitagawa, Mr. Masaaki Murakami, Mr. Hiroto Nakayama, Ms. Noriko Tsuda and Mr. Daisuke Yanase. Finally, we would like to thank Prof. Junjiro Kanamori, the Director of IIAS, for his dignified opening speech.

Organizers Akira Ohbuchi(Chief) Kazuhiro Konno Atsushi Moriwaki Noboru Nakayama Sampei Usui

V

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Introduction to Arakelov Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shu Kawaguchi, Atsushi Moriwaki and Kazuhiko Yamaki

1

Double Covering of Smooth Algebraic Curves . . . . . . . . . . . . . . . . . . . . . Changho Keem

75

Algebraic Surfaces with Quotient Singularities - Including Some Discussion on Automorphisms and Fundamental Groups . . . . . . . . . . . . . 113 JongHae K e u m and De-Qi Zhang Linear Series of Irregular Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jungkai A . Chen and Christopher D. Hacon

143

Hecke Curves on the Moduli Space of Vector Bundles . . . . . . . . . . . . . . . 155 Jun-Muk Hwang Minimal Resolution via Grobner Basis . . . . . . . . . . . . . . . . . . . . . . . . . . Yukari It0

165

Deformation Theory of Smoothable Semi Log Canonical Surfaces . . . . . . . 175 Yongnam Lee

........................

187

On the Asymptotic Behavior of Admissible Variations of Mixed Hodge Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gregory Pearlstein

205

Modular Curves and Some Related Issues Viet NguyenKhac

Degeneration of SL(n)-Bundles on a Reducible Curve . . . . . . . . . . . . . . . 229 Xiaotao Sun Refined Brill-Noether Locus and Non-Abelian Zeta Functions for Eliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lin Weng

245

List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263

vii

INTRODUCTION TO ARAKELOV GEOMETRY SHU KAWAGUCHI, ATSUSHI MORIWAKI AND KAZUHIKO YAMAKI

2

SHU KAWAGUCHI, ATSUSHI MORIWAKI AND KAZUHIKO YAMAKI CONTENT

Introduction 1. Arithmetic Chow Group 1.1. Introduction 1.2. Currents 1.3. Arithmetic Variety and Arithmetic Chow Group 1.4. Arithmetic Intersection Theory 1.5. Arithmetic Chow Group and Push-forward of Arithmetic Cycles 1.6. Height of Arithmetic Variety 2. Arithmetic Riemann-Roch Theorem 2.1. Characteristic Forms 2.2. The Bott-Chern Secondary Characteristic Class 2.3. Arithmetic characteristic class 2.4. Analytic torsion and Quillen metric 2.5. Arithmetic Riemann-Roch Theorem 3. Existence of a Small Section 3.1. Small Section 3.2. Arithmetic Euler Characteristic 3.3. Arithmetic Hilbert-Samuel Theorem and the Existence of a Small Section 3.4. The comparison of LP-norm and sup-norm 3.5. Proof of Theorem 3.3.3 3.6. Proof of the arithemtic Hilbert-Samuel theorem 4. Adelic metric and admissible metric 4.1. Adelic metric and intersection number 4.2. Admissible metric and cubic metric 5. Arithmetic height function 5.1. Definition of arithmetic height functions and their properties 5.2. Height functions on abelian varieties 5.3. Adelic metric and height function 5.4. Intersection number of nef Coo-hermitian line bundles 5.5. Relation of height function and intersection number 6. Bogomolov's conjecture 6.1. Equidistribution theorem 6.2. The proof of Bogomolov's conjecture 7. A generalization of conjectures of Bogomolov and Lang 7.1. Statement of Bogomolov plus Lang 7.2. Small points with respect to a group of finite rank 7.3. Proof of Bogomolov plus Lang Appendix references

3 3 3 6 11 13 15 16 18 18 20 21 22 25 25 26 26 28 31 33 35 42 42 47 51 51 53 54 55 57 59 59 61 63 63 64 69 71 73

INTRODUCTION T O ARAKELOV GEOMETRY

3

This note was first written in Japanese for intensive lectures of Arakelov geometry organized by Moriwaki from December 8 to December 10, 1998 at Kyoto University. These lectures were intended to give an overview of Arakelov geometry and a proof of Bogomolov's conjecture for general algebraic geometers. From that time, we have considered that this note should be available for not only Japanese but also a broader range of readers. We, however, had no chance to translate it into English. Fortunately, during the meeting "Algebraic Geometry in East Asia," Professor Usui recommended its translation. Here we would like to express hearty thanks for his proposal. The final goal of this note is a generalization of conjectures of Bogomolov and Lang. For this purpose, in the first part, we introduce "Arithmetic Chow group" and "Arithmetic Riemann-Roch theorem," in which we do not give a rigorous proof for every result, but we believe that this is a good introduction of Arakelov geometry. In the middle part, we consider "Existence of a small section," "Adelic metric and admissible pairing" and "Arithmetic height function,'' in which several techniques of Arakelov geometry are used. In the final part, we give a proof of Bogomolov's conjecture and a generalization of conjectures of Bogomolov and Lang. Here we would like to explain a generalization of conjectures of Bogomolov and Lang in the case of a curve and its Jacobian. Let K be a number field, X a geometrically irreducible projective curve of genus greater than or equal to 2 over K , and J the Jacobian of X . Let us fix an embedding L : x(Z)-+ J ( Z )and a NQron-Tate pairing ( , ) : J ( Z )x J(R)-+R. Let r be a subgroup of J ( K )with dimq r @ Q < co. Let (r@R)' be the orthogonal complement of r @ R in J(R)@ R in terms of the NQron-Tate pairing. Let Lr : x(R)+ (I? @ R ) be~ the compositions of maps

-

x(R) ---L J(Z)

projection

J(R)@R (F@R)'. Then, a generalization of conjectures of Bogomolov and Lang says that the fiber of Lr is finite and the image of Lr is a discrete subset of ( ~ B R ) 'in terms of the metric arising from the NQron-Tate pairing. If we consider the case I? = J ( K ) ,then the first assertion is nothing more than Mordell's conjecture. Moreover, if we consider the case r = {0), then we have Bogomolov's conjecture. $1, $2, $3 and $7 were written by Kawaguchi, $4 by Yamaki, and $5 and $6 by Moriwaki. We hope that this note will be useful for anyone who wants to know Arakelov geometry.

1.1. Introduction. In the Arakelov geometry, one considers, roughly speaking,

4

SHU KAWAGUCHI, ATSUSHI MORIWAKI AND KAZUHIKO YAMAKI

schemes over Z with "infinity," instead of algebraic varieties over a field, vector bundles with "metrics at infinity," instead of vector bundles. Szpiro [34] wrote about the Arakelov geometry that Put metrics at infinity on vector bundles and you will have a geometric intuition of compact varieties to help you. Let us see this analogy by comparing a compact Riemann surface with Spec@). Let X be a compact Riemann surface. (A) Let f be a nonzero rational function on X . Then we have

It follows from the residue formula that

) Div(X)/Rat(X). Set Rat(X) = {div(f) ( f E (C(X)*) c Div(X), and C H ~ ( X = Then the map deg : Div(X) -+ Z induces the map deg : CH'(X) -+ Z. (B) Let L be a holomorphic line bundle over X , and s a nonzero rational section of L. Put div(s) = up(s) . b] E CH' (x).

x

PEX

Then, as an element of CH'(X), div(s) depends only on L, i.e., it is independent of the choice of s. (C) Let Pic(X) be the set of isomorphism classes of holomorphic line bundles over X . Then, we have the isomorphism where s is any nonzero rational section of L. In particular, through this isomorphism, deg : Pic(X) -+ Z is defined. Next, we consider X = Spec(%). (A') Since we consider a scheme at "infinity," let us set

Let f E Q be a nonzero rational number. We set

S ( f )=

up(f). [p]

+ u,(f).

[m] E

%(x).

INTRODUCTION T O ARAKELOV GEOMETRY h

h

We define deg : Div(X) -+ R by

It follows from the product formula that h

h

deg(div(f)) =

C

vp(f) logp

1 + ;;v,(f)

= 0.

(In this case, the product formula is an obvious consequence of the prime factorization.) Set

the map deg CH (X)

h

h

and S 1 ( x ) = G ( x ) / ~ ~ ( x )Then . the map deg : Div(X) :

-1

+ Z.

-

Z induces

(B') Let C be a line bundle over X . Since we consider a L'metricat infinity," let us take a hermitian metric h : LC x La: ---t C on Cc = C @, @. We denote the pair (C, h) by line bundle. Let s be a nonzero rational section of C. Put &(s) =

v,(s).

+ (-logh(sc,sc))

z, and call it a hermitian [m] E Z 1 ( x ) .

p:prime

Then, as an element of S 1 ( x ) , &(s) depends only on L, i.e., it is independent of the choice of s. (C') Two hermitian line bundles = (C1,hl) and = ((12, h2) are said to be isomorphic if there exist an isomorphism $ : L1 4 C2 of line bundles such that the induced map $c : (Clc, hl) -+ (Lac, h2) is an isometry. Let %(x) be the set of isomorphism classes of hermitian line bundles over X. Then

z2

Zl : %(x)

-

4

S 1 ( x ) , C Hg

( S )

is an isomorphism, where s is any nonzero rational section of C (cf. Proposition 1.3.4). In particular, through this isomorphism, deg : Pic(X) --, Z is defined. h

h

To sum up, by adding "infinity" to Spec@) and considering hermitian line bundles over Spec(Z), one has the degree map for Spec@), similar to the degree map deg for a compact Riemann surface, in the sense that deg(div(f)) = 0 for f E Q \ (0).

&

-

A

6


Such an analogy between the ring of integers of a number field and a compact Riemann surface have been noted by many mathematicians such as Hasse and Weil to name a few. Arakelov [2] generalized this analogy to a 2-dimensional case, and established the intersection theory on arithmetic surfaces, which corresponds to that on projective surfaces over @. Then, Faltings proved in [5] among other things a Riemann-Roch theorem on arithmetic surfaces. Gillet and Soul6 (cf. [9] [ll],[3], 1121, [13]) developed its higher dimensional theory, including arithmetic cycles and their intersections on arithmetic varieties, arithmetic Chern classes of hermitian vector bundles, and an arithmetic Riemann-Roch theorem. Such theory of arithmetic varieties is called the Arakelov geometry. We remark that [32] is a good reference of the Arakelov geometry and [33] is a good quick guide to it. Let us consider a case of dimension 2 2. We set

+

X = Proj (z[X, Y, Z ] / ( y 2 Z = x3 x z 2 ) ) . This is an example of arithmetic surfaces (cf. 51.3). To consider "infinity" means to consider the compact Riemann surface Xc = Proj (C[X, Y, z ] / ( Y ~ z - x3- x z 2 ) ) . Moreover, to consider a line bundle with a "metric at infinity" means to consider a pair = (C, h), where C is a line bundle over X and h is a hermitian metric on LC. Then, what is %(x)? The case X = Spec(Z) kept in mind, it would be natural (undoubtedly with hindsight) to think that (div(s),- 1% h(sc, s c ) ) becomes an "arithmetic divisor" on X , where s is a nonzero section of C. In what follows, we will give the precise definitions of arithmetic varieties, arithmetic divisors, the arithmetic Chow groups on an arithmetic variety etc., due to Gillet and Soul6. In fact, an arithmetic divisor on an arithmetic variety X is a pair ( 2 , g ) such that 2 is a cycle on X and g is a "Green current" on X(C); And (div(s),- log h(sc, sC)) above is indeed an arithmetic divisor on X. So, let us first define Green currents in the next subsection. 1.2. Currents. Let X be a d-dimensional compact complex manifold. Let AplQ(X) be the space of Cw differential forms of type (p, q) on X. We endow Ap>q(X)with the compact Cw topology: Namely, a sequence {%) converges to ,7 in Aplq(X) if and only if (1) there exists a compact set K such that for any n the support of % is contained in K and (2) any order derivation of 7, uniformly converges to the corresponding derivation of qrn

Definition 1.2.1. We call a continuous linear functional T : ~ d - p , d - Q(XI @ +

7


a current of type (p, q) on X . Let Dp>q(X)be the set of currents of type (p, q) on X.

Example 1.2.2. For w E AP>q(X),set

Then, [w] E Dp,q(X). Via a subspace of Dplq( X ).

[.I

-

: ApvQ(X)

DPlq(X), w

H

[w], Aplq(X) is regarded a s

Example 1.2.3. Let w be a differential form of type (p, q) on X with locally integrable coefficients. Then, in the same way as in Example 1.2.2, one obtains the current [w] E DplQ(X). Example 1.2.4. Let X be a non-singular projective variety over @ and Y a subvariety of X of codimension p. Then, we have the Dirac type current by E Dplp(X) defined by

-

where Yns is the set of non-singular points of Y. Note that, if n : ? Y is a n*(q) holds, and thus resolution of singularities of Y, then the equality by(7)) = Jyn, 77 converges.

SF

Example 1.2.5. More generally, let Y = C , n,Y, (n, E Z)be a cycle of codimension p on X . Then, we have the current by Dp)"(X) defined by by = C , nabye. -

A current T E DpJ'(X)

is said to be real if T(7) = T(q) for any example, 6y as above is a real current. Let us define differential operators on d T E Dp+'>q(X) and 3~E D P > ~ l (x) + by

7)

E ApJ'(X). For

ep,, Dplq(X). For T E Dp?Q(X),we define

We see from the Stokes theorem that [dw] = d[w] and [awl = 3[w] for w E Ap?q(X). Moreover, we set d=d+3,

Note that ddc = 9 8 5 . For w E AP>q(X),we similarly have [dw]= d[w] and [dCw]= dc[w]. The pull-back of differential forms induces the push-forward of currents. Indeed, let n : X --+ Y be a holomorphic map of compact complex manifolds and

8


g an element of Dp,q(X). Then, the push-forward of g, which is an element of D ~ X-dim+ ~ X-dim ~ ~ and is denoted by T, (g), is defined by n,(g)(rl) = g(r*rl). Y7q+dim

Definition 1.2.6 (Green Current). Let X be a compact Kahler manifold and Z c X a cycle of codimension p. A Green current for Z is a current g E DP-'J-'(X) such that there exists w E ApJ'(X) with ddc(g)

+ 6z = [w].

Let X be a compact Kahler manifold and L a line bundle over X . A Coo-hermitian metric h on L is a Coo-fieldof hermitian inner products in the fibers of L. Namely, for each x E X , h, : L, x L, ---+ C is a hermitian inner product, and h, is Coo with respect to x. We call := (L, h) a Coo-hermitian line bundle .

z

Example 1.2.7. Let X be a smooth projective variety over C, E = (L, h) a Cwhermitian line bundle over X , and s a nonzero rational section of L. Then, since - log h(s, s) is locally integrable, [- log h(s, s)] defines a current in D O l O ( ~(cf. ) Example 1.2.3). The following PoincarBLelong formula shows that [- log h(s, s)] is actually a Green current for div(s). Theorem 1.2.8 (PoincarBLelong formula). Let X be a smooth projective variety over C, = (L, h) a Coo-hermitian line bundle over X , and s a nonzero rational section of L. Let q ( z ) E A1"(X) be the first Chern form o f z . Then, the following formula holds in D'>~(x):

z

(1.2.8.1)

ddC[- log h(s, s)]

+ 6div(s)= [cl(z)].

Proof: Let d be the dimension of X . Step 1 The assertion holds if the support of div(s) is a normal crossing divisor. Indeed, for any p E X , one can take an open neighborhood U of p and local coordinates 21,. . . , zd of U such that Supp(div(s)) is locally defined by zlz2. . . zk = 0. By the partition of unity and the linearity, it suffices t o show that, for any E ~ d - 1 , d - 1 (U) with compact support,

We will show this equality in the appendix (cf. Lemma A.l). Step 2 We treat a general case. Set D = div(s). By Hironaka's theorem 1171, there exists a proper morphism n : 2 + X such that (i) 2 is smooth, (ii) E = T*(D),,~is a normal crossing divisor,

INTRODUCTION TO ARAKELOV GEOMETRY

(iii) nIh',Supp(E): 2 \ Supp(E) -t X

\ Supp(D) is isomorphic.

On the other hand, we have -

log h(s, s ) ddc7 =

Jx +

S,

cl (E) A q =

We write E = E', where dim D. Thus, we have

D

- log r*h(n*s,n*s) ddc(x*7) CI (

n * ~A) r * ~ .

is the strict transform of D. Then dimr,(E1)

(TIy)d'mY~+l1 h(xlz)(Y)= (dim yQ+ 1) degL, (YQ)' where degL, (YQ) = cl ( L Q ) ~Y~' ~ n IyQ]. Also, let Y be a subvariety of Xo with dimY 2 1. Let y be the Zariski closure of Y in X. Then, we define the height of Y with respect to ( X , z ) by

Remark 1.6.1. The value -I-hi(X) is also often called the height of X [K : Ql When dimY = 0, i.e., Y is a maximal point y, we define the height of y with respect to (X,Z)by

where A, is the Zariski closure of y in X. Note that, if p : normalization of A,, then we have deg

zy A,

(zlA,)= & (p'(Z)).

-t

C

X is the

The height of Y with respect to ( X , z ) has played an important role in Ullmo and Zhang's proof of the Bogomolov conjecture. Indeed, roughly speaking, Theorem 5.5.1 says that inf{the height of sufficiently general point of XQ)

2 the height of X

> inf{the height of point of XQ).

Further, the equidistribution theorem (Theorem 6.1.1) is applied when these three terms coincide. To obtain a "nice" height of Y, in 54.1, we take various models (X, and consider the intersection as a limit of models.

z)

2.1. Characteristic Forms. In this subsection, we review characteristic forms without proof. Let X be a complex manifold of dimension d. Let E be a holomorphic vector bundle of rank r ober X . We denote by AO(x) (or by C m ( X ) )the vector space


19

of C m functions on X. Moreover, we denote by Ap74(E) (or by Ap,4(X, E ) ) the space of C" forms of type (p,q) on X with values in E. Set An(E) = @p+q=nAp3q(E). A connection V in E is a C-linear map

v : AO(E)

A'(E)

-+

such that, for any s E AO(E)and f E A O ( x ) , V ( f s ) = df 8 s

+ fVs.

According to the decomposition A1(E) = A1lO(E)@ A0,l(E), we have the decomposition V = V110 @ VO". We extend a connection V to the covariant exterior differential

v : A ~ ( E -+ )

--tk+l(E)

by the Leibnitz rule V(QB S ) = de B s + (-1)"

A AS

(QE A ~ ( x ) ,s E AO(E)).

The curvature of V is defined to be

v2: AO(E) + A ~ ( E ) . Then, V2 is Ao(x)-linear. Hence, V2 is an element of A2(E 8 E V ) . A Cm-hermitian metric h on E is a Cm-field of hermitian inner products in the fibers of E (cf. 91.2). Namely, for each x E X , h, : Ex x Ex --+ @ is a hermitian inner product, and h, is C m with respect to x. We call E = ( E l h) a Cm-hermitian vector bundle . A hermitian metric h determines a natural connection in E as follows.

Lemma 2.1.1. Let = ( E l h) be a Cm-hermitian vector bundle. Then, there exists a unique connection VF with the following properties:

: = BE, where BE is the Cauchy-Riemann operator; (i) V (ii) VE is unitary, i.e., d h(s, t ) = h(Vgs, t )

+ h(s, V g t ) .

Since V 0EL2 = -2d E = 0 and by the unitarity V? (1,l)-form with values in E 8 E V .

= 0, the curvature

is a

Let 4 E Q[[Tl,.. . ,T,]] be a symmetric formal power series in r-variables and 4(k)the k-th part of 4. Let A be a commutative Q-algebra and Mn(A) the set of (n, n)-matrices with coefficients in A. Then, there exists a unique map

dk) : Mn(A) + A

20


such that (1) @ ( k ) ( d i a g ( ~.l.,,A,)) . = 4 ( k ) ( ~ .1. ,.,A,), and (2) for any P E GLT(A) and M E MT(A),@ ( k ) ( ~ ~ = ~ @("(M). - l ) For a Coo-hermitian vector bundle = ( E , h) over X, we define the differential form 4(B) by

z

The form (denoted also by +(E,h)) is called the characteristic form of E associated to 4. The form 4(E, h) is d-closed. If h' is another hermitian metric of E, then +(E,h) 4(E, h') is d-exact. Thus, as a cohomology class, 4 ( F ) does not depend on the choice of h, although, as a Coodifferential form, 4(E, h) depends on h. Some examples of

are

k t h Chern form: ck = Clp(X)be the morphisms defined i n 51.3.

z

Theorem 2.3.1 (Arithmetic Characteristic Class). Let 4 E Q [ [ T l , . . ,T,]] be a symmetric formal power series in r variables. Then, for each regular arithmetic variety X and Cm-hermitian vector bundle = ((E, h ) of rank r over X , one can uniquely associate $(El E @ E I p ( x ) @E ~20 with following properties: ( i ) For any rnorphism f : X + y of regular arithmetic varieties, f*($(Z)) =

a

$(f*@));

(ii) If ( E , h ) = ( L ,h i ) $ ... @ ( C ,h,), then (iii) u(&Z)) = ~ ( E c ) ; (iv) If E : 0 -+ S + 4 vector bundles, then

z a

-+

0 is a short exact sequence of CM-hermitian

22


+

(v) Write $(TI T, . . . , T, hermitian line bundle,

+ T ) = CiZo & ( T i , . . . ,T,.)Ti;

The class $(F) is called the arithmetic characteristic class of

Then, for any C m -

z associated to 4.

We refer to 112, 54.11 and 132, IV Theorem 31 for its proof. Notice that, taking w in (iv) and using (iii) and w a = ddc, we get 4(& @ Gc) - 4(zc) = ddc(F(Ec)). This is one of the properties of the Bott-Chern secondary characteristic (cf. The(iii) becomes orem 2.2.1 (ii)). We remark that for a Cm-hermitian line bundle w(Z1( z ) ) = cl (&). This is the Poincar6-Lelong formula. By taking ck, ch, and td as 4, we obtain the arithmetic lc-th Chern class zk(z), the arithmetic Chern character & ( z ) , and the arithmetic Todd class &I@), respectively.

z,

2.4. Analytic torsion and Quillen metric. Let X be a compact Kahler manifold of dimension d with Kahler form 0. Let E be a holomorphic vector bundle. The determinant of cohomology of E is the one-dimensional vector space d

X(E) = @ (det H q ( X ,E))(-')~ q=o Let h be a Cm-hermitian metric on E. In what follows, we will define two metrics (called the L2-metric and the Quillen metric) on X(E). Let A0+7(x,E ) be the vector space of C m sections of the vector bundle of I \ ~ ( T * ( O ~ ~@) X E.) For z E X , let ( ), be the hermitian product on I \ q ( ~ * ( O l @E l)~) associated with w and h. Then the normalized L2-hermitian metric ( , )LZis defined by

Let d : A0>q-'(X, E ) -t A034(X,E ) be the Dolbeaut operator and d* : -A094(x, E) + A0>q-l(x,E ) its adjoint operator with respect to ( , )L2. Let 0, = ad* 3*d : A0>q(x,E ) + A0>4(x,E ) be the B ~ a ~ l a c i a n . Set 7-14(X, E ) = Ker Elq. This is the vector space of harmonic (0, q) forms with values in E. By Hodge theory, W ( X , E ) is isomorphic to H Q ( X ,E). Since W ( X , E) inherits the L2-metric from that on A034(x,E ) , it induces a metric hHq(X,E) on Hq(X, E ) by the above isomorphism. The hermitian metric on X(E), induced from W ( X , E)'s is denoted by hLz and called the L2-metric. Next, we will define the Quillen metric on X(E). Let u(Uq) be the set of eigenvalues of 0, and Eq(X) the eigenspace of A'?~(X,E ) with eigenvalue X E ~ ( 0 ~For ) .

+


sE

(C

with Re(s) >> 0, set C4(s) =

C X€40,)\{0)

dim E4 (A) AS .

It is known that Cq(s) converges absolutely for Re(s) >> 0, extends meromorphically to the whole complex plane, and is regular a t s = 0.

Definition 2.4.1 (Analytic torsion). The real number T ( E , h) = C(-1)4"9 0 with the following property: for any X with [XI < Xo, Xddc(f) + c l ( z ) is semipositive in some neighborhood of x. When M is a manifold, semipositivity (resp. positivity) coincides with that in the ordinary sense. Let f : X -+ Spec@) be a projective arithmetic variety. Let = (13, h) be a Cw-hermitian line bundle over X. We define to be vertically nef if it satisfies the following conditions:

z

z

~


29

(i) For any one-dimensional integral scheme C such that f ( C ) is zero-dimensional, deg(Clc) 0 (Nef along the fibers); (ii) cl ( z ) is semipositive.

>

Moreover, we define

to be vertically ample if it satisfies the following conditions:

(i) L: is f-ample; (ii) cl ( z ) is positive. The purpose of this section is to prove the following theorem and its corollary.

Theorem 3.3.1 (Arithmetic Hilbert-Samuel Theorem). Let f : X -+ Spec@) be a and be Coo-hermitian line bundles over X . projective arithmetic variety. Let Assume the following:

n

:

(i) CQ,is ample over XQ,; (ii) is vertically nef.

Then, we have A

xsup (X,

- d+l

4 zanc3 n')= d e g(d( z+l (I)!

,d+l

+

+d+l)

Corollary 3.3.2 (Existence of a small section). Let X be a projective arithmetic variety and a Coo-hermitian line bundle over X . Assume the following: (i) CQ,is ample XQ,; (ii) is vertically nef; (iii) &(?I (z)d+l) > 0

z

z

Then, for any suficiently large n,

has a nonzero small section.

In order to prove Theorem 3.3.1, we first prove the following theorem.

Theorem 3.3.3. Let f : X -+Spec(Z) be a projective arithmetic variety. Let be Coo-hermitian line bundles over X . Assume the following: and (i) XQ, is regular; (ii) C is vertically ample. Then, we have

Remark 3.3.4. For the proof of Theorem 3.3.3, we use the arithmetic RiemannRoch theorem. We remark that, Abbes and Bouche [I] gave a comparatively short proof of Theorem 3.3.3 (in the case N = Ox)that does not use the arithmetic Riemann-Roch theorem.

30


Following [32], we will prove Theorem 3.3.3 in four steps:

Step 1: We compare xSup(X,Pn 8g) and

XL2

(x,

pn8n)(53.4);

Step 2: We relate X L ~ ( X , Z 8@X~ ) to S ( & ( d e t R f , ( P n 8 N),hQ)) (53.5); Step 3: We state the asymptotic behavior of analytic torsions, which appear as the the difference of L2-metrics and Quillen metrics in Step 2 (53.5);

-

Step 4: We relate &(?I (det R f, ( c @ 8 ~ N), hQ)) and deg(Q (c)d+l)n d + ~ using , (d l ) ! the arithmetic Riemann-Roch theorem (53.5).

+

In 53.6, following [37], we will prove Theorem 3.3.1, by taking a generic resolution of singularities of X and using Theorem 3.3.3. In the sequel, we will see that Corollary 3.3.2 follows from Theorem 3.3.1 and Minkowski's convex body theorem. Proof of Corollary 3.3.2 (Assuming Theorem 3.3.1): Set I?, = H O ( X CBn) , and Vn = H O ( X CBn) , 8~R. We endow Vn with the sup-norm induced by P n . By Minkowski's convex body theorem (Theorem 3.2.2), we find (dim Vn) log X l (I?,)

(3.3.4)

< (dim Vn) log 2 + log volsUp(I?,).

On the other hand, by Theorem 3.3.1, we get

Since CQ is ample, we have, for sufficiently large n, dim Vn = d! Thus, we obtain (dim Vn) log 2

+ log volsUp(rn)

=(dim Vn) log 2 - xsup(rn)

Hence, if

(z)d+l) > 0, then

(3.3.4)

(dim Vn) log 2

+ log v ~ l , ~ ~ ( I ' ~- m)

( n -+ m ) .

It follows from (3.3.4) and (3.3.4) that, for sufficiently large n , (3.3.4)

log X l (I?,) < 0.


31

m,

Since, by definition, Xl(rn) = min{llsllsupI s E s # 01, (3.3.4) shows that there < 1. Therefore, for sufficiently large n, exists a nonzero element s E rnwith C@nhas a nonzero small section. 3.4. The comparison of P - n o r m a n d sup-norm. Let U C C be an open unit disk. Let 4(z) be a holomorphic function on U that extends continuously to Then, for any p > 0, we have

u.

0. If F is rn-regular, then, for any n m, (1) F is n-regular and (2) H O ( X E'@L@")@HO(x, , L) -+ H O ( X F@L@("+')) , is surjective (cf. [19, I1 $1 Proposition 11).

>

Lemma 3.6.2. Let k be a field and .rr : Y -t X a morphism of projective varieties over k . Let L be an ample line bundle over X and M an ample line bundle on Y . Then, @ H'(Y, x*(L)@"@ M @ ~ ) a,b>O

is a finitely generated k-algebra.

36


Proof: Step 1: Suppose L and M are very ample. Set d = dimY. Let bo be a sufficiently large integer. Then, for any j, 0 5 j 5 d and q, 1 5 q 5 d,

H q (Y,n*(L)"

(3.6.2.1)

@ M @ ( * - ~ ) )= 0.

This equality shows that, for fixed j, r * ( L ) @ is j bo-regular. Then, for any b

> bo,

Thus, for any b 2 bo, X, ~

~ ( ~ @- n, 4 () M

@~)

y, n * ( ~ ) @ ( ~@- qMBb )

This equality shows that T , ( M @ ~is) d-regular. Then, by [19, I1 $1 Proposition 11, for any a 2 d, we have

H0 (x, L@' &I r . ( ~ @8~HO) )(X, L )

-

H0

(x,L N a f '1 8 n . ( ~ ) @,~ )

and hence, we have

).

HO(Y, T * ( L ) @ B ~M @ ~B) HO(Y, n * ( ~ + ) ) HO(x,n * ( ~ ) @ ( ~B+Ml )@ ~ In particular, for any a 2 d, we have (3.6.2.2) H0 (Y, n* ( L ) @@~M m b ) &I H 0 (Y, n* ( L ) @ ( ~ -+ ~ )H)0

(x,c*( L ) @&I~M @ ~ )

On the other hand, substituting j = d in (3.6.2.1), we get

This equality shows that T * ( L ) @is~bo-regular. Then, for any b 2 bo, (3.6.2.3) H' (Y,T * ( L ) @g~M @ ~ o ) HO (Y, M @ ( ~ - ~ o+) HO (x,n * ( ~ )B@M~@ ~. )

)

Combining (3.6.2.2) and (3.6.2.3), we find that, for any a HO

(x,r * ( ~ ) M@@~~ o )HO

2 d and b 2 bo,

(Y,n * ( ~ ) @ ( ~8-HO~ ) (Y, ) M'(~-~O)

+ H0 (X, n*(L)@"@ M @ ~. ) This gives the assertion. ~ both very ample. By a Step 2: Take A and B such that LBA and M @ are similar argument in Step 1, for each 0 5 i < A and 0 5 j < B,


is a finitely generated k-algebra. Then,

is also a finitely generated k-algebra.

r:

Let X b e a projective arithmetic variety, and = ( C , 11 . 1 1 ) a CM-hermitian line bundle over X . W e give the sup-norm on H O ( x C , ) @z R. As in $3.2, let X i ( H O ( xC , ) ) b e the successive minima o f H O ( x C , ) for 1 < i rk H O ( x C , ). W e denote X,~HO(X,L)(HO(X, C ) ) by Xmax(H0(X,C ) .

0).

Proof: It suffices t o show that H q ( Y ,T * ( L ) @ 8 ~M@" @ H ) = 0

( q > 0) for sufficiently large n and N . W e may assume that L and M are very ample. Set d = dim Y . Take a sufficientlylarge no. T h e n , for any j, 0 5 j 5 d and q, 1 5 q 5 d , we have H' (Y,T * ( L ) @ @~ M@("'-') @ H ) = 0. T h u s , n * ( L ) @ jis no-regular for each j. T h e n , for any n

and hence, for any n

> no,

> no,

This shows that n,(M@" @ H ) is d-regular. Thus, for any N

T h e n , for any N

> d,

> d and n 2 no,

Proof of Theorem 3.3.1:

For simplicity, we assume N = Ox.

( i ) W e fix a generic resolution o f singularities o f X , which we denote by n : 2 + X ( c f . 91.6). Also, we fix a Cm-hermitian line bundle M over 2 such that M is very ample and t h e first Chern form c l ( M c ) is everywhere positive. Moreover, we fix a nonzero section sl o f ~ ' ( M 2 ),. Let C i be t h e sup-norm o f sl. Set C 1 = m a x ( C i , 1 ) . (ii) W e fix a sufficiently large a such that

Since HO

(zQ, T * ( L Q ) @8 a MQ1)

= H0

(2, T*(L)@ @~ M - I ) @Z Q,

39


there exists a nonzero section s2 of H0

(2,T * ( C ) @8~M - I ) .

Let C; be the

sup-norm of sg. Set C2 = max(C4,l). (iii) For any x E XQ(C) and a function 4 defined over .rr-l(x), we set \I+\\ = sup,,,-i(,) l4(y) 1. Then, the coherent sheaf T, ( 0 2 ) over X is equipped with norm. Hence, the coherent sheaf 3 := Xomo, (7r,(02), Ox) is also equipped with norm. We fix an integer b >> 1 with H O HO

(x,FQ@I c&)= H O

(XI3 8 cb) BZ0, there exists a nonzero section s3

of I f 0 (X, 3@J Cb). Let C$ be the sup-norm of s3. Set C3 = max(C$, 1). -(iv) Let CL be the constant in Lemma 3.6.3 for .rr : 2 -, X and (C, M ) . We set C4 = max(Ci, 2). First, we sketch the proof. In the rest of this subsection, we write additively the tensor product of line bundles. (For example, we write C M for C 8 M . ) The theorem is proven from the following: For any positive number 6,

+

In order to show (3.6.4.1), we take sufficiently large N depending on 6 , and $ M . Roughly speaking, one can show (3.6.4.1) by comparconsider n* (C)

+

xsUp(2,n(rr*(Z) + f r ~ ) and ) xSup (X, n z ) rem 3.3.3 to estimate xSup( 2 , n ( * * ( ~+ ) $37)). ing

More precisely, since .rr*(C)

+

+ AM ..

for n

-

m, and by using Theo-

is a Q-line bundle, we multiply it by N , i.e.,

+ m)+

we compare xsUp(X, ( n N i)T) and xmP(2,n ( ~ x * ( ~ ) r * ( ~for ) )i (0 5 i 5 N - 1). We remark that, in the following, N is fixed (according to c) and only n goes to 00.

Let N , n be positive integers, and i an integer with 0 _< i 5 N N>a+b. Step 1: Multiplying by sy, one has a morphism

a

:

rl := H O

(X, ( N n

+ i)C)

-

-

1. Assume

r2:= H O (2,( N n + i)**(C) + n M )

Put Vl = I'l @JzB and V2 = l72 8~R. Then, Vl and V2 become normed spaces with the norm induced by the sup-norm of Z and M.The norm of a is bounded from above by C;. Step 2: Multiplying by s;, one has a morphism HO

(2,{ ( N - a - b)n + i)n*(C) + n M )

-

HO

(2,{(N - b)n + i)n*(C))

40


Also, multiplying by s;, one has a morphism

H0 (2,{(N - b)n

+ i}n*(C)) = H 0 (X, { ( N

-

b)n + i}C

+s*(ox))

-

H0 (X, ( N n

+ i)C) .

Let ,O be the composite of these two morphisms: ,O : r3= H 0 (2,{ ( N - a

-

b)n

+ i}a*(C) + n

~

)

rl = H0(X, ( N n + i)C) .

Put V3 = r3@Z R. Then, V3 becomes a normed space with the norm induced by the sup-norm of E and a . The norm of ,O is bounded from above by CTCF. Step 3: We will bound xsup(rl) from above. We equip cr(V1) c Vz with the induced norm. Since a is injective, we have

Applying Lemma 3.6.1 to (V,I?) := (Vz, r z ) and (V', I") := (a(Vl),( ~ ( r l ) )we , have

+ (rk(r2) - rk(r1)) ( N n + N + n ) log (&I). Let 7: 2 -+ Spec@) be the structural morphism. Since ( N n + i)n*(C) + n M is ?-ample and c l ( ( N n + i)a*(z) + n a ) is everywhere positive, we can apply T h e e rem 3.3.3 to ( N n + i)a*(C)+ n M and we get ~ s u p ( ~ ( r 15) )xsup(r2)+log (rk(rz)!)

-

(Nn (d

+ i)d+ldeg(?1 (Eld+') + o ( N ~ ~ ~ + + 'ON) (nd+l).

+ I)!

Here, 0 ( ~ ~ n ~means + l ) that

I

O K s 1 )

1 is bounded from above by a constant

I wI

that is independent of N , n , i. And oN(nd+l)means that, if we fin N , then I converges to 0 as n goes to infinity. On the other hand, by Lemma 3.6.4, for any n >> 1, N >> 1, and 0 5 i 5 N - 1, we + ~M8" ) @ H ) = 0 (Vq > 0). Then, by the Riemann-Roch have Hq(Y,T * ( L ) @ ( ~ " @ theorem for algebraic varieties, we find I

nd

=d! deg

+

+

+

( ( ( ~ n i)** (LC) M ~ ) ~ oN ) (nd-l)

+ i)d deg(C;) + o ( N ~ - ' ~ +~ orv(nd-I), ) d!

- (Nn


and rk(r1) = dimQ(Vl) nd d! ( N n i)d d e g ( ( ~ c ) d ) O(Nd-'nd) d!

+

= - d e g ( ( ~ ~ c ) d oN(nd-') )

+

+

+ oN(nd-').

Therefore, we get

n r k ( r l ) log C1 = O(Ndndfl )

+ oN(nd)

log (rk(r2)!)= O(Ndnd)log(O(Ndnd))= oN(ndf ') (rk(r2) - r k ( r l ) ) ( N n

c 4 + N + n ) log = O(Ndndf') + oN(nd). 2

Combining these estimates, we obtain

+

-

( N n i)d+l deg(6 ( z ) d f l ) O(Ndndf') oN(ndf l ) . (d l ) ! Step 4: We will bound x s u p ( r l ) from below. We equip P(V3) c Vl with the induced norm. Since P is injective, we have xsup(r'1) 5

+

+

+

+

xsup(r3) i xsup(PF3)) nrk(r3) logC2C3 Applying Lemma 3.6.1 t o (V, r ) := (K, r l ) and (V', I") := (P(V3),P(r3)),we have

Thus we find xsup(rl) 1 xsup(r3)- n r k ( r 3 ) logC2C3 - log (rk(Fl)!)

+ (rk(r1)

-

rk(r3)) ( N n

+ N + n ) log (:-C4 ) .

By a similar argument as in Step 3, it follows from Theorem 3.3.3 that

+

-

(Nn z)~+' deg(21(z)d+') O(Ndndfl) oN(ndf l ) . (d l ) ! For rk(r3), we have a similar estimate as for r k ( r l ) and rk(F2): For any n >> 1, N >> 1, and 0 5 i N - 1, we have xsUp(rs) =

+

+

+

0 ) and let L1,. . . , Ld line bundles with a integrable adelic metric on X . Let Ci,n))n=1,2,... , d i , n and BiYn be those in Definition 4.1.4 for El. For an (rill . . . , nd)E Pld, let Xnl ,,,,,? be a model of X dominating any Xiyni (i = 1 , 2 , . . . , d ) . We denote the pullback of Ci,ni to Xnl,,,,,n,by the same symbol Ci,ni.

-

Theorem 4.1.5. Under the above situation, we have the following. ( 1 ) cnl ,...,nd := deg(21(Ll,nl). . . 2 1 ( L d v n d ) ) converges as m i n { n l , . . . ,n d ) + cm, and the limit does not depend on the choice of sequences of models. W e denote it by cl ( E l ) . . . cl ( E d ) . ( 2 ) cl ( L 1 ). . . cl ( E d ) is multilinear with respect to E l , . . . , Ed. Proof: Let us take two d-tuples ( n l ,. . . ,n d ) (ni, , . . . , n&)E Rid. Let X be a model and Bi be the pull-backs dominating two models Xn ,,,..,nd and X,; ,...,, . Let Ci, and Bi be the pull-backs of f i , d i p , and Bi,ni to X respectively, andd let J?:, and Bi,,: to X respectively. Then ( X ,Li) and ( X I13;) are models of of Ci,n:, respectively. We shall (X, L i ) and induce the same metrics Li on as Li,ni and prove

4

l&(?l(Ll)

' '

'cl(ld)) - &(zl(ci) 'cl(L&)) 1 +0 ' '

as min{nl, . . . , n d , ni, . . . , n:) + co. By the definition of convergence of the models, we can see the following.

45


(1) There exists a non-empty open subset U of Spec(OK) such that 11. IILi,%,p = 11 . IILi ,*,,p for any i = 1 , . . . ,d, for any n and n', and for any finite place P E U. (2) For any positive number E, if nk and n; are sufficiently large, we have

1

log 1 , < { c l o g # ( ~ K / ~ if) v is a finite place P, II . IILk," E if v is an infinite place,

for any v E MK. Note that there are only finitely many places a t which log

ll~ll~~,~ survive. (They

E

and c;@"* are line are outside of U.) Let ek be a positive integer such that lfek bundles and coincide with

LF* on the generic fiber X.

Let s k be a rational section of goal is to estimate

ffek

@

Then

Ix

(c;@'*)

=

Ox.

( L ~ @ " * )of- ~which restriction t o X is 1. Our

-

1 Ik:= deg(?1(f;). . . ~ l ( t ; - l ) ? l ( f k + l ) . . 4 ( f d ) -div(sk)), ek

namely, ~~(?l(cl)...?l(cd) )G ( ? l ( c ; ) . . . ~ l ( G ) ) l

For each a = (al, . . . ,ak-1, ak+l,. . . , ad) E (0, bundles M i ( a ) ' s as follows: for i with 1 5 i 5 k - 1, M i ( a ) := and for i with k

A', if ai = 0,

B: i f a i = l ,

+ 1 5 i 5 d,

Note that each M i ( a ) is vertically nef. Now we put

It is straightforward that d

d

we define hermitian Q-line

46


We shall examine how much each place contributes to j k ( a ) . If P is a finite place, then # ( O K / ~ ) - '< ~ ~IIsk(x)llP < # ( O K / ~ ) ' ~ ~ for any x E X ( K ) . By the definition of the metric associated with a model, this tek[Xp] are implies both DP,l := [div(sk)]p cek[Xp] and Dp,2 := -[div(sk)]p effective divisors, where [Xp] is the divisor of fiber over P and [div(sk)]pis the part of div(sk) supported in Xp. Since each Mi(a)is vertically nef, we have

+

+

for j = 1,2. This implies that the absolute value of the contribution of P to jk(a) is not greater than t ( l o g # ( O ~ / P ) ) c i ( M i ( a ) ). . . c i(Mk-~(a))ci(Mk+i(a)) . . . ci(Md(a)), where Mj(a) is the restriction of M l ( a ) to the generic fiber. By the assumption K , (L3i,n)K)izl,...,d,n~ is bounded in NS(XK)@ R, there exists a positive that {(di,n) number C independent of the choice of (nl, . . . ,nd) or (ni, . . . ,n i ) such that c l ( M ~ ( a ) .) . ci(Mk-l(a))ci(Mk+l(a))...ci(Md(a)) 5 C. Consequently, the absolute value of the contribution of a finite place P ( 4 U ) to jk(a)is not beyond 6 log #(OK/P)C. Next let us consider the contribution of infinite places. The contribution of an infinite place a to J k ( a ) is

where ci(.) denotes the curvature form. Taking account that all the curvature forms which appear above are positive semi-definite and that I log IIskll,,l < Eek, we can similarly observe that its absolute value is bounded with and we conclude that it is not greater than tC. Accordingly, we have

and thus we obtain I&(El(Ll). . .El(&))

-

G(EI(~~ . .Ei(Eb)) ).

1

47


which implies the convergence. Let c denote this value. If {(X;,,, z;,,)), is another sequence of models, then we can obtain another c'. If we consider the third one

(Xi,ll .Ci,l), (X;,l,L i , l ) ,

xi,^, z i , ~(X;,2, ) , J C , ~(Xi,3r ) , .C.,2 3 ) ,...,

we have the third c". We must have c = c" and c' = c", hence we conclude c = c', which completes the proof of ( 1 ) . ( 2 ) is immediate from the definition.

4.2. Admissible metric and cubic metric. Let X be a projective variety over an algebraically closed field F and L = (L, 11 . 1 1 ) be a line bundle on X with a continuous and bounded metric. Suppose a surjective endomorphism f : X + X and an isomorphism 4 : L @1 ~ f * L (d > 1 ) are given. We define a metric 11 . 11, on L as follows: 1

I I . 111 = I I . I I , Il . lln = (4*f*Il. 1171-1)'. Then we have the following theorem.

( 1 ) (11 . 11n)n=1,2,...converges u n i f o m l y to a metric, say 11 . 110. ( 2 ) 1 1 . [lo is the only continuous and bounded metric with 1 1 . Ilo = ( 4 * f * l l .110) lld . (3) If we replace 4 by Ad, 11 . 110 is replaced by ( ~ l ~ l ( ~. 110. -~)I1

Theorem 4.2.1.

Proof:

(1) Put

h := log -.I I . 112

I I . 111

Then we have

II = II II. .11n-1

n-2

( 4 )

(h)

and hence

%

, log converges uniformly Since 11 ( ( l / d ) f f * ) (h) llsup = ( l / d k )11 h ~ ~ s ua psequence II. I to a function ho on X ( F ) . Therefore, I I . I ( , converges uniformly to 1 1 . Ilo := eholl. [I1. ( 2 ) jF'rom the construction, we see 11 . \lo is a required continuous and bounded metric. Let (1. ((6be another continuous and bounded metric satisfying the conditions. Then g := log is a continuous and bounded function on X ( F ) and satisfies

m

( ( l / d ) ff *) ( g ) = g. Therefore Ilgllsup = (l/d)llgllsup,and since d 11g11sup = 0.

> 1,

we have

48


(3) Let all . [lo be the metric satisfying the condition (I) with respect to XI$. Let denote the metric f * 11 . 110 on f * L. Then for any x E X (F)and a local section 1 around x, we have

11 . [lo,

which implies a = (a1A[) 'Id. Thus we have the conclusion. Let X be a projective variety over a number field K and L an ample line bundle on X. Suppose that we are given a surjective morphism f : X --t X and isomorphism I$ : L @2 ~ f * L (d > 1). Since L is ample, f is a finite morphism of degree ddimX. Fix a model (X, C) over SpecOK of (X, L) such that C is vertically nef. Let 11 . 11 be the metric on L associated with (X,C), and let e be a natural number such that C@"is a line bundle and L@elx = L@". Then, there exist a non-empty open subscheme U C SpecOK, a surjective morphism f u : Xu -, Xu and an isomorphism 4;" : (c@")@~ f;(Cge). By virtue of the above theorem, we obtain for each place v a metric 11 Ilu,o with 11. Ilu,o = (6f * l . ~ l ~ On ~ the) other ~ ~ hand, ~ .we can see from . 1lp)'ld for each P E U \ ((0)). Therefore, the construction that 11 . lip = (ff*ll the sequence of adelic metrics defined by

(1 . 1 ,

with 11 . adelic metric 1) . is vertically nef. Indeed, let morphism

X be the normalization of the composite

=

(6f*ll . ~ l , - l ) " ~converges to an (I$* f*ll . ~ l ~ ) " We ~ . claim that this metric 1 . /lo

f, : X,

+

z

=

'

X" xu X and put En := i;C. Then the metric Il.lln is nothing more than the metric associated with the model (x,,E:('~~")), and &:('ldn) is vertically nef. Thus, each I/ . 11, is vertically nef and so is (L, 1) . 110). If X is an abelian variety, then we have the n-times morphism. Using this, let us define the cubic metric (up to constant) on any line bundle as follows. Let A be an abelian variety and let L be a line bundle on A. We define morphisms s1,2,3, ~ 1 , 2 S2,3, , s3,l and si (i = 1,2,3) from A x A x A to A by ~ 1 , 2 , 3 ( ~ 1 , ~= 2 ,x1 ~ 3+) x2 + 23, ~ i , j ( x l , x~~ 3, =) X i 3- X j , si(xl,x2, 53) = xi. Now we put Cub(L) := ~;,~,~(L)@S;,~(L@(-~))@S;>~(L@(-~))@~;,~ (L@(-l))@s;(L)@Sf(L)@s;(L).


49

Then, the cubic lemma gives us an isomorphism $L : O A x A x A-' Cub(L). If L is equipped with a metric 11 . 11, we can canonically endow a metric Cub(L) = Cub((L, 11 . 11)) on Cub(L) by pulling back the metric, and we further have a metric A4~. 11 ' On O A X A Xvia

Definition 4.2.2. Let 1., I be the canonical metric on O A x A x A . We call a continuous and bounded metric (1 . 11 on L a cubic metric if there exists a positive number c such that 11 . 114, = cl ., , I Note that it is independent of the choice of $L. This notion appears in [22] in case of archimedean valuation. We have a basic result on cubic metrics.

Proposition 4.2.3. Let L be a line bundle on an abelian variety A. (1) There exists a cubic metric on L. (2) If 1 1 . 111 and 11.112 are cubic metrics on L, then there exists a positive number c such that 11 . 111 = cJ1. 112. Proof: (1) First we suppose L is even, i.e., [-l]*L r L. Fix an isomorphism $ : [2]*L-+ LB4. Let 11 . 1) be the admissible metric with respect to [2] : A + A and $. Then, we see [2]*Cub(L)= [2]*( s ; , ~ , ~ ( L @)s;,~(L@-')8 s;,~(L@-') @ S;,~(L@-') 8 s; (L) @ s;(L) 8 s$(L)) = s ; , ~ , ~ [ ~ ] *@( L ~ ;), ~ [ 2 ] * ( L @8 - l~) ; , ~ [ 2 ] * ( L @8- ls:,~ ) [2]*(L@p1)

8 s;[2]*(L) 8 s;[2]*(L) 8 sj[2]*(L)) s ; , ~ , ~ ( L8 @s~; ,) ~ ( L - ~@) S ; , ~ ( L - ~@) s ; , J ( ~ - ~ ) @ s ; ( ~ @@~s;(Lm4) ) 8 s;(LB4)) =~ub(L)@~.

This implies the metric of Cub(L) is the admissible metric with respect to

and

where $' is the canonical isomorphism $' : [2]*L-' L@-4 induced by $. The metric of Cub(L) is also regarded as an admissible metric on C ? A X A X Avia $L, which must be cl ., I Thus we see that 11 . 11 is a cubic metric on L. If L is odd, i.e., [-l]*L 2 L-l, then we can construct a cubic metric in the same way using [2]*L r L ~ . -+

50


For a general M , we can find an even line bundle Ml and an odd one M2 such that M = MI @ M2. Taking account that C u b ( ~ 11 )8 C u b ( ~ 2= ) Cub(~l@ Mz), we can easily see that the metric on M defined as a product of a cubic metric on MI and that on M2 is a cubic metric. (2) It is enough to show that if the metric of C U ~ ( O A 11.11), is the canonical metric up to constant, then so is 11 . 11. For a global section 1 of UA, we define a function X : A(F) -+ R by x log l ( l ( ( ( x )Since . 11 . 11 is continuous and bounded, so is A. By the assumption that the metric of Cub(OA,11.11) is the canonical one up to constant, we have

-

and hence, the map from A x A to R defined by comes out to be Z-bilinear. Taking account that it is also continuous and bounded, we must have X(x+y) -X(x) - X(y) +X(O) s 1. This implies that the function

-

is Z-linear, and by continuity and boundedness again, we have X(x) X(0) and hence I1 . 11 = X(0)l . Ic,,. Note that the above proof of (1) tells us that the admissible metric with respect to a given isomorphism 11, : [n]*L% LBn is a cubic metric. Finally, let us consider the case of F = @. Let A be an abelian variety over @. Let .rr : @g -t A be the universal covering of A compatible with the natural group law. Let us fix the coordinates zl, . . . , zg of @g. Then the 1-forms dzl, . . . , dzg,dzl, . . . , d% are naturally regarded as 1-forms on A. If h is a Cm-hermitian metric on a line l , , , .ci,j , ~ = Ei j ,and a 1-form q such that bundle L, there exist a matrix ( ~ ~ , ~ ) i , j =with (;,j-dzi

cl(L, h) =

A dzj

+dv

i,j

By ddc-lemma, there exists an R-valued Cm-function f such that ciVjG d z i A dzj

c1(L, h) =

+ ddcf .

ij

Hence if we put hl = ef h, we have ci,jG d z i A dzj.

cl (L, hi) = i ,j

It is straightforward to see that hl is a cubic metric. Moreover, we see any cubic metric is necessarily Cm-class, and if L is ample, then the curvature form of its cubic


51

metric is positive definite. In particular, if L is in addition even, the admissible metric with respect to an isomorphism [2]*L 2 L @is~ Cm-class and its curvature form is positive definite.

5.1. Definition of arithmetic height functions and their properties. First of all, let us recall the definition of height functions. Let K be a number field and OK the ring of integers of K . We denote by K(@) the set of all embeddings K into (C. Let C be a line bundle on S p e c ( O ~ )i.e., , C is a projective and finitely generated OK-module of rank 1. For a E K(@),the tensor product C 80, @ in terms of the embedding a : OK --, @ is denoted by C,. We give a hermitian metric 1) . 1 , of L, for each a E K(@). The collection {(I . Il,),EK(c) is called a hermitian metric of C and is simply denoted by 11 . 11. Moreover, a pair (C, 11 . 1))of a line bundle C and a hermitian metric 11 . 11 of C is called a hermitian line bundle on Spec(OK) and is often denoted by For s E L: \ {0), let us consider the following: log llsI1, E R. log #(C/sC) -

z.

UEK(C)

Then, by virtue of the product formula, - _ the above number does not depend on the choice of s, so that it is denoted by deg(C). Let X be an integral projective scheme over Spec(OK). For x E XK(K), let A, be the closure of the image Spec(X) -+ XK -+ X. Let be a continuous hermitian line bundle on X. Namely, for each a E K(@), we endow a continuous hermitian metric to the line bundle C, = C 8 @ on Xu = X 8 (C as before. Then, we set

z

More precisely, letting

L, be the normalization of A,, G ( ( x , s ) I ~-,~is) given by

& ( ( x , i f ) l A z ) . We call this the height of x with respect to (x,Z)

(cf. 51.6). Here we consider the following proposition. For simplicity, we set X = XK and L = CK.

Proposition 5.1.1. Let Bs(L) be the base locus of L, i.e., Bs(L) = Supp ( c o k e r ( ~ O (L) ~ ,8 Ox --, L)) . Then, there is a constant C such that h(,,z)(x) 2 C for all x E (X \ Bs(L))(E). Proof: We set

Let {sl, . . . , sl) be a system of generators of H'(x, L) as OK-module. a = max { s u ~ l l s i l l ~ ) . lsisl

UEK(C)

52


For x E ( X \ BS(L))(R),we can find a generator si with si(x) # 0. This means that

Therefore, we can see h(X,Z)(.) 2 - [K : Ql log(a).

Corollary 5.1.2. If L for all x E X ( T ) .

2

Ox,then there is a constant C such that Ih(x,z)(x)(5 C

Proof: Apply Proposition 5.'1.1 to

z and L

-8-1

.

Recall that a model of (XIL) is a pair ( ~ , zsuch ) that X is an integral projective ) the generic fiber XI and is a continuous hermitian scheme flat over S p e c ( 0 ~with = LBn for some positive integer n (cf. 54.1). line bundle on X with L 8 n ( ~

z

Corollary 5.1.3. Let (x, a constant C such that

z)and (x', z')

be two models of (XIL). Then, there is

Ih(x,c)(XI - h(xl,zl)(x) l L C for all x E X ( T ) . Proof: Note that X is birational to XI. Let XI' be the graph of the birational correspondence X --+ X' in X x XI. Let p : X" -+ X and p' : X" + X' be the natural morphisms. Then, it is easy to see

On the other hand, p*(z) coincides with (p')*(zl) on the generic fiber XI1 Spec(OK). Thus, Corollary 5.1.2, there is a constant C such that

-+

Hence we get our corollary. ) ) set of all (resp. Let us denote by F U ~ ( X ( K ) , R )(resp. B F U ~ ( X ( ~ ? ; ) , Rthe bounded) real valued functions on ~ ( x )By . the above corollary, the class of h(,,,) in F u n ( x ( K ) , R ) / B F u n ( X ( R ) , R) does not depend on the choice of the model ( ~ , z of) (X, L), so that the class of h(x,z) is denoted by h(X,L)or hL. The following Northcott's theorem is very important (for details, see Lang [21] or Serre [31]).

53


Theorem 5.1.4. If L is ample, then the set { x E X ( K ) 1 [ K ( x ): K ] I dl ~ L ( x < ) M) is finite for all M and d. Here by abuse of notation, h~ is a representative of the class h L . 5.2. Height functions on abelian varieties. Let K be a number field, A an abelian variety over K , and L a line bundle on A. Let h~ be a height function associated to ( X IL ) . Let us consider a problem to find a good representative of the class of hL. By virtue of the cubic theorem and Corollary 5.1.2, we can see that

+ y + z ) - ~ L ( +X Y )

~ L ( X

-

+

~ L ( Y 2) - h

~ ( +z 2 ) + ~ L ( x+) ~

+h ~ ( z )

L(Y)

is a bounded function on A(K)x A@) x A ( K ) . The following lemma is well known (for example, see [21]).

Lemma 5.2.1. Let G be an abelian group, and f

:G

-t

R a function on G. If

f(x+y+z) -f(x+y) -f(y+z) -f(z+x)+f(x)+f(y)+f(z) is a bounded function on G , then there exists a quadric form q and a linear form 1 uniquely such that +O(1). f ( X I = 4 ( x )+ By the above lemma, there are a quadric form q~ and a linear form lL on A@) such that (mod B F u n ( A ( K ) ,R)). h~ = q~ 1~

+

+

q~ 1~ is denoted by AL, and is called the canonical height function (or Ne'rol Tate height function). If L is symmetric (i.e., [ - l ] * ( L )= L ) , then lL = 0. Thus, in this case, k L is a quadric form. Finally, let us consider the following proposition. Proposition 5.2.2. W e assume that L is symmetric and ample. Then we have the following. ( 1 ) For all z E A@), k L ( x ) > 0. ( 2 ) For x E A@), & L ( x )= 0 if and only i f x E A(T)t,,. Proof: ( 1 ) By the definition of height functions, i L B n = n h L Thus, we may assume that L is generated by global sections. Therefore, by Proposition 5.1.1, there is a constant C such that k L ( x ) C for all x E A ( K ) . Here, since k L is a quadric form, for all n > 0 ,

>

n 2 i L ( x )= i L ( n x )> C. Therefore, taking n -+ m, we obtain L L ( x )

> 0.

( 2 ) Since iLis a quadric form, it is obvious that if x E A(R)~,,, then h L ( x )= 0. Conversely, we assume that h L ( x ) = 0. Let us consider a subgroup ( x ) generated

54


by x. Note that if x E A(K1) for a finite extension field K1, then (x) A(K1). Moreover, kL(nx) = n2kL(x) = 0 for a11 n. Thus, by Northcott's theorem (cf. Theorem 5.1.4), (x) is a finite group. Therefore, x E A(K)~,,.

5.3. Adelic metric and height function. Let X be a projective variety over K, and L a line bundle. Let 11 . (1 be an adelic metric of L. We assume that the adelic metric ( X , z ) is given by the limit of a sequence of continuous models {(Xn,zn)). Here X, is a projective arithmetic variety over S p e c ( 0 ~ whose ) generic fiber is X. Moreover, is a continuous hermitian Q-line bundle on X, which gives rise to L on the generic fiber. Then, we have the following proposition.

z,

Proposition 5.3.1. For any positive 6, there is a positive number N such that, for all n , m 2 N and all x E

~(x),

I~(,,L)(X) - h(xm,z,)(x)l 5 In particular, If we set h(x,z)(x) = limn,,

6

h(Xn,zn)(x),then

Proof: Its proof is essentially the same as that of Theorem 4.1.5. For reader's convenience, we dare to give it. First of all, there is a non-empty Zariski open set U of Spec(OK) such that all (Xn, 13,) are same over U . We set S = Spec(OK) \ U. For P E S, the metric over P induced by the model (X,, 13,) is denoted by 1) . lip,,. Since #(S) < a, for any positive number 6, there is N such that, for all n, m N and all P E S,

>

I

Sup log ( l ~-:l ~ ~ : ) ~ < ~ l ~ ~ # ( o ~ / p ) ~ Moreover, taking a larger N if necessarily, we may assume that for all n, m all a E K (@),

2 N and

I (,!:I;:::) 1

sup log ---

5 €.

>

For n, m with n , m N , let y be the graph of the birational correspondence X, --+ Moreover, let x, : y -+ X, and x, : y -, Xm be the natural Xm in X, x X,. morphisms. Let s be a rational section of x:(Ln) @ ~ ~ ( 1 3 , such ~ ) that s gives rise to 1 over U . Then, for each P E S and each a E K(@),we have sup 11% Ilsll~l5 ~ l o g # ( o K / P ) and

sup 11% IIsIIcrI 5

6.

jFrom the above observation, we can easily see that the contribution to evaluate


55

over P is bounded by c[K(x) : K] log #(OK/P), which is the intersection number of A, and €(the fiber of y over P ) . Moreover, the contribution over o is bounded by c[K(x) : K]. Therefore,

+ CPES log #(OK/P), then

Thus, if we set B = [K : Q]

lh(xn,z,,)(x)- h(xm,z,,,)(x)l 5 Therefore, we get our proposition.

5.4. Intersection n u m b e r of nef Cm-hermitian line bundles. Let X be a be a Cm-hermitian line projective arithmetic variety with d = dim XQ. Let bundle on X. As in 53.3, we say is vertically nef if C is relatively nef with respect to X -4 Spec(Z) and the Chern form cl(Z) is semi-positive. Moreover, Z is said to be horizontally nef if for any 1-dimensional subscheme C flat over Spec(%), _ deg (Clc) 2 0. Further, we say is nef if is horizontally and vertically nef. Then, we have the following theorem.

z

z

z

h

z

-

T h e o r e m 5.4.1. Let E l , . . . , Cd+1 be nef CM-hermitian line bundles on X. Then

& ( & ( Z l ) .. .&(Zd+l)) 2 0. Proof:

Let us begin with the following lemma.

L e m m a 5.4.2. Let Z be a nef Cm-hermitian line bundle on X, and M a vertically nef Cm-hermitian line bundle such that for all integral subscheme r p a t over Z,

& ( s ~ ( A ~ , ) ~ ~ ~ (>~ O.Q ) + ~ ) Then, for all 0

5 i 5 d + 1,

& (8(ZP . h( ~ ) ~ + l - '2) 0 Proof: We prove it by induction of d. If d = 0, the our assertion is obvious. Here we assume that d > 0. First, let us consider the case where 0 5 i < d 1. --@n Since ( E ~ ( M ) ~ +> ' ) 0, replacing (n > 0), we may assume that by M there is a small section s of M, namely, s E H'(x, M ) \ (0) with JJsJJSup < 1. Let div(s) = a l r l . . . a,r, be a decomposition as a cycle. Here a j > 0. Then,

+

&

+ +

56


Thus it is positive by the hypothesis of induction. Next let us consider the case where i = d 1. For a rational n ~ b e t,r we set Ct = t n . Let us consider a polynomial P ( t ) given by P ( t ) = deg (21(zt)d+1). Here we claim the following.

+

z+

Claim 5.4.2.1. If t > 0 and P ( t ) > 0, then P ( t ) >_ td+'&

(21(TGi)d+1).

First of all, by the hypothesis of induction and the assumption P ( t ) > 0, we get

for all integral subscheme I? flat over Z.Therefore, in the same as before, we can see that (21(2).21(zt)d) 2 0. Then

Thus we get our claim. Let us go back to the proof of the lemma. We set to = max{t E R I P ( t ) = 0). We assume to > 0. Then, by the above claim, for all t > to, ~ ( t2) td+l& Therefore, taking t

( Z ~ ( M ). ~ + ~ )

+ to,

0 = P(t0) >_ t ; + l G (21(x)d+l) > 0.

This is a contradiction, that is, to 5 0. In particular, P(0) lemma.

> 0.

Hence, we get the

Let us go back to the proof of the theorem. We also prove this by induction of d. If d = 0, then our assertion is obvious, so that we assume that d > 0. Let us fix a vertically nef Coo-hermitian line bundle M such that, for all integral subscheme r flat over Z,

5( q A l r ) d i m ( r ~ ) + l For a positive rational number t, by using the above lemma,

& ((4(&+I) + tz1( A ) ) ~ + ' ) > 0.

57


+

Therefore, if we take a large number n , n(C t M ) has a small section s. Let div(s) = a l r l . . . be a decomposition as cycle. Here a j > 0. Then,

+ + sere

Thus, by the hypothesis of induction, we can see

& (ZI(z1)

' '

'2l(zd) 'zl(zd+l

+ t z l ( m ) ) ) 2 0.

Thus we get our theorem because t is an arbitrary positive rational number.

5.5. Relation of height function and intersection number. Here let X be a projective arithmetic variety with d = dim XQ.

Theorem 5.5.1. Let Z be a Cm-hermitian line bundle on X such that nef and CQ is ample. Then, we have the following inequalities:

{

inf sup YSXQ x€(XQ\Y)(@ h(x.z)(x) Proof:

}

zis vertically

- d+l &(""' 2 inf- h(x,z)(x) (d 1)deg(C&) X€X(Q)

+


Lemma 5.5.2. ~f &(E~(z)~+') > 0, then

z

Proof: By our assumption, replacing by -@n C (n > 0), by Corollary 3.3.2 we may assume C has a small section s. We set Y = div(s). Then, for any x E (XQ \ Y)(Q),

- Let us start the proof of the theorem. Let c be a real number with 0 < c < 1. Let A be a hermitian line bundle on Spec@) given by 71 = (Ospec(z), cl . I). First, let us

consider the left inequality. Let X be a rational number with

58


Then, by an easy calculation,

G ((6(c)

-

~4(a*( 2 1 ) ) ~ >~ 0, ' )

where 7r : X -+ Spec(Z) is the natural morphism. Therefore, noting that is vertically nef and ample on XQ, by the above lemma, we can see

Thus,

{

inf h(,,,(z) sup YSXQ xg(x~\Y)(G!) Hence, we get the left inequality.

1

z - AT*@)

--

2 Adeg(A).

Next let us consider the right inequality. Let p be a rational number with

Then,

This means that rem 5.4.1,

i n f h(x,z-,,*(;i,,(x) 2 0. xEX(Q) - pn*(A) is horizontally nef, so that it is nef. Thus, by Theo-

G ( ( 6 (z)p ~( al * ( ~ ) ) ) ~ +>' )0. -

Hence

& (&(T)~+') > p(d + 1) d e g ( ~ $ d e g ( ~ ) . h _

Thus, we get the right inequality.

Corollary 5.5.3. Let X be a projective variety over Q and bundle on X. Then

an ample adelic line

Proof: Let (X,, En) be a sequence of Cm-models which gives rise to the adelic line bundle 1. Here is vertically nef, so that we may assume that is vertically nef on X,. Thus, by the previous theorem,

zn


Here h

_

lim & ( E ~ ( Z ~ ) ~ +=' )d e g ( ~ d i l ) and 71-03

lim h ( x n , r ) ( ~ ) = h ( x , z ) ( ~ ) -

n-m

Note that the left hand is a uniform convergence, namely, for any positive is a number n such that

t,

there

for all x E X ( 0 ) . By using this, we can see lim

n+m

i n h xEX(Q)

n (x) =

inf- h(x,E)(x) z€X(Q)

and

Thus, we get our corollary. 6. BOGOMOLOV'S CONJECTURE 6.1. Equidistribution theorem. Let K be a number field, and X a geometrically irreducible projective variety over K . For an embedding a : K + @, let Xu = X @ K @ in terms of the embedding a. Let a : K -+ (I: be an extension of (T : K + C. Using a, we have a natural map x(K) + Xu(@). For x E ~ ( r )let, O(x) be the orbit of x by the action Gal(E/K). We denote by O,(x) the image of O(x) by the map a : X ( R ) + Xu(@). Note that O,(x) does not depend on the choice of the extension a. Let {xm)zXl be a sequence in X ( R ) . We say {x,):=~ is generic if every subsequence of { ~ , ) z = ~is Zariski dense in X . - Let L be an ample line bundle on X and 11 . 11 an adelic metric of L such that L = (L, 11 . 11) is vertically nef, that is, 11 . 11 is given by a sequence of vertically nef l . we have the following theorem. models { ( X n , ~ n ) ) ~ =Then,

Theorem 6.1.1 (cf. [35], [36], [39]). Let { ~ , ) z = ~be a generic sequence in x(K). We assume the following. (1) c l ( L ) is positive on Xu. (For positivity, see $3.3.) (2) h(x,z,(x) > 0 for all x E X ( x ) . (3) limrn-rn h(X,t) (xm) = 0. Then, as currents on Xu, we have a weak convergence

60


Namely, for any Cm-function f on Xu,

Proof: Let f be a Cm-function on Xu. Clearly, we may assume that f is real as follows. On Xu, we valued. For a real number A, we change the metric of change it by exp(-Xf)ll. 1) and on the other component of X(C), we keep the metric. Then, since c l ( L ) is positive and Xu is compact, there is a positive number Xo such that, for any X with [XI Xo, Ex is vertically nef. Thus, by Corollary 5.5.3,

zx

0, that is, i L ( x ) = 0. Thus, x E A(K)~,. This indicates the theorem in this case. For this purpose, we assume that dimX > 0. Since G ( X ) = {0), by the previous : Xm + a m ( X m ) is birational. lemma, if we take a large integer m, then amJXm By abuse of notation, we denote am(x,by a,. In order t o prove our assertion, we may replace K by a finite extension field of K , so that we may assume that X is defined over K and X m + a m ( X m )is birational over K . Let )I . / I be the admissible metric induced by [2] : A + A. Then, E = (L, 1) . 1)) is vertically nef and cl(x) is a positive Cm-form. Moreover, h(A,L)= hL. Let pi : Am -+ A be the projection to the i-th factor and Ern = p;(T) @ . . . @ p&(z). The metric of Em is an admissible metric with respect to [2] : Am -+ Am. Moreover, note that

62


In the same way as before, we define L-1on Am-'. Since the set of all proper subvarieties of X is countable, we denote it by {Yn)r=l. Then, by our assumption, for each n , we can find xn E X ( X ) with xn $2 Uy=l Y , and hL(xn) 5 l l n . Then, the sequence {x,) is generic and limn,, LL(xn) = 0. Let N be the set of all natural numbers and let us fix a bijection /3 : N -+ Wm. Denoting the i-th entry of P(n) by Pi(n), we set x(n) = ( x ~ , ( ~ . . ). , x ~ ~ ( ~Thus, ) ) . we get a sequence {x(n)) in X m . Then, limn,, h(xm,.t;m,(x(n)) = 0. Moreover, {x(n)) is Zariski dense in X m . Indeed, note that { ~ ( nI )n E N) = {(xe,, . . . ,xem) I ( e l , . . . ,ern) E Nm). Thus, we can see by induction of m that { (xe,, . . . ,xe,) I (el, . . . , em) E Nm) is Zariski dense in X m . Here let { Z i ) F l be the set of all proper subvarieties of X m . We choose x(ni) with x(ni) $ ~ j 2,. = Then, ~ {x(ni)) is a generic sequence and limi+, h(Xm,L,)(x(ni))= 0. Here since am: X m + a m ( X m ) is birational over K, there is a Zariski open set U of a m ( X m ) over K such that am : a k l ( U ) 1U. Considering a subsequence of { ~ ( n i ) )we , may assume that x(ni) E a ; ' ( ~ ) . Let us consider {am(x(ni))). This is generic on a m ( X m ) and

Therefore, if we fix an embedding 0 : K then, by Theorem 6.1.1, we have

-+ (C and

we set d = dim X m = dim a m ( X m ) ,

on X T , and

-

on am(Xm),. Thus, each convergence holds on a , ' ( ~ ) , since a i l ( ~ ) , U,, we have

and U,, respectively. Here

63


on a;'(U),. Moreover, each side is a Coo-form. Thus, the equality holds on X F . On the other hand, a, : X m -+ a m ( X m )is not an isomorphism on the diagram {(x, . . . , x) ( x E X}. Thus, the right hand is not a positive form on the diagram. This is a contradiction because the left hand side is positive. Finally, let us consider a general case. Let A' = A/G(X) and n : A -, A/G(X) be the natural morphism. Moreover, we set X' = n(X). Then, G(X1) = (0) and n-'(XI) = X . Let L' be a symmetric ample line bundle on A'. Choose a positive integer a such that LBa 8 nX(L)@-Iis ample, which implies n*(LLl) 5 aLL. Then, for every positive E , {XI E X1(K) 1 kv(xl) 5 E} is Zariski dense. Indeed,

Therefore, by the previous observation, there is a x' E A'(x)t, Thus, X = n-'(x'). Hence we can easily get our theorem.

with

X' = {x').

In this section, we discuss a generalization of conjectures of Bogomolov and Lang, which Poonen [26] and Zhang [40] independently proved and which Moriwaki [25] extended for finitely generated fields.

7.1. Statement of Bogomolov plus Lang. Let K be a number field. Let A be an abelian variety over K and L a symmetric ample line bundle on A. As in 55.2, let ^hL : A@) -+ IR be the Nhron-Tate height function associated with L. For x, y E A(E),put

xL

1 (x, y)L = - @L(X 2

+ y) - ~ L ( x-) hL(?)) . A

: A ( E ) x A ( E ) + IW is a symmetric bilinear form, which is is quadric, ( , Since called the Ne'ron- Tate height pairing associated with L. By Proposition 5.2.2, ( , ) gives a hermitian inner product on A(x)/A(R)t,. For X I , . . . ,xl E A(X), we set

~ L ( X.I. ,. , X I ) = det ((xi, xj)L) Let r be a subgroup of finite rank in ~ ( z )i.e., , dimQ(r 8 Q) < oo. Note that

r is a subgroup of A@)

with

rdiv = {x E A(K) I nx E r for some positive integer n ) is also a subgroup of finite rank in A ( x ) . The following theorem is a generalization of the conjectures of Bogomolov and Lang.

64


Theorem 7.1.1 ([26], [40]). Let the notation be as above. Let X be a subvariety of AW Fix 71,. . . , yn E r such that they form a basis of I? 8 Q.Assume that, for any positive number E > 0 , is Zaristi dense i n X . Then, X is a translation of an abelian subvariety of AK by an element of r d i v . If we take r = 0 in the above theorem, then we have Bogomolov's conjecture (Theorem 6.2.1). As we will see below, Theorem 7.1.1 is also a generalization of the following Lang conjecture, which Falting proved. (For its proof, we refer to [7], [8].)

Theorem 7.1.2 (Lang's conjecture over a number field). Let K be a number field, and A an abelian variety over K . Let X be a subvariety of AT. Then, there exist abelian subvarieties C1, . . . , Cn of Ag and yl, . . . ,yn E x ( K ) such that

where X ( K ) is the Zaristi closure of X ( K ) . In the following subsections, following [25] (which underlines Poonen's idea and uses a geometric trick), we will prove Theorem 7.1.1. Its proof is based on Bogomolov's conjecture and Lang's conjecture. In the meanwhile, let us check that Theorem 7.1.1 is indeed a generalization of Lang's conjecture. Namely, assuming Theorem 7.1.1, let us show Theorem 7.1.2. If dim(X(K)) = 0, we are done. Let Y be a positive dimensional irreducible component of X ( K ) . Then Y(K) is Zariski dense in Y. On the other hand, Mordell-Weil's theorem says that A(K) is a finitely generated group. Then, applying Theorem 7.1.1 to Y and r = A(K), we find that Y is a translation of an abelian subvariety. Thus, we get Theorem 7.1.2. 7.2. Small points with respect to a group of finite rank. As in $7.1, let K be a number field, A an abelian variety over K , L a symmetric ample line bundle on A, and I' a subgroup of finite rank in A(R). In this subsection, we assume that there exists a subgroup roof r with ro A(K) and ro8 Q = r 8 Q.Note that, if we replace K by a suitable finite extension field, then there always exists such a ro. A nonempty subset S A ( T ) is said to be small with respect to r if there exists a decomposition s = y(s) z(s) for each s E S with the following properties: (i) y ( s ) E r for every s E S ; (ii) For any E > 0, there is a finite proper subset M of S such that ( z ( s ) z, ( s ) ) ~5 E for all s E S \ M. (Warning: M is proper, i.e., S \ M # 0. )

c

+


65

A nonempty subset S A ( K ) is said to be small if S is small with respect to (0). For example, if S is a finite set, then S is small if and only if S contains a torsion point. Let us state Bogomolov's conjecture (Theorem 6.2.1) in terms of small sets:

Theorem 7.2.1. Let S C A@) be small. Then, there are abelian subvarieties C1, . . . , C,, torsion points cl, . . . ,c,, and finite nontorsion points bl, . . . , b, such that -

T

S = U ( ~ i + ~ i ) ~ { b l ,,b .m. )., i=l

where 3 is the Zariski closure of S in A. Let X be the positive dimensional irreducible component of 3.Put S' = Then, S' is small. By Theorem 6.2.1, X is a translation of an abelian subvariety by a torsion point. Proof:

S

n X.

Let T be a subset of A@), and F a finite extension field of K in K . For x E A(R), set OF(x) = { ~ ( xE) A ( z ) I a E G a l ( r / ~ ) ) For . n 1 2, let Pn : An + An-' be the homomorphism defined by Pn(xl,. . . ,x,) = (x2 - X I , 2 3 - X I , .. . ,xn - xl). Then, we define the subset Vn(T, F ) of A@)"-' by

F), denote the Zariski closure of Vn(T, F ) in An-'. Let B n ( ~ Notice that, if T' is a subset of T and F' is a finite extension field of F, then Vn (TI, F') Vn (T,F ) and thus Bn(TI, F') Bn(T, F). Let [N] : A + A be the homomorphism given by x + Nx. Then, since [N](Vn(T,F ) ) = Dn([N](T), F),we have B,([NI(T), F ) = [N](Dn(T, F)).

c

c

We say that a pair (T, F ) is n-minimized if the following conditions are satisfied: (0) T is an infinite set. (1) For any infinite subset T' of T and finite extension field F' of F, B,(T', F') = Dn (T, F ) ; (2) For any positive integer N , [N](%(T, F ) ) = D n ( ~F). , (This condition is equivalent to B,([N](T), F ) = D,(T, F ) .) Before we state Lemma 7.2.2, let us make a few remarks. Let f : A --t A' be a homomorphism of abelian varieties over R . Let L' be a symmetric ample line bundle on A. Since L is ample, we can take a positive integer a such that Lga 8 f *(L1)-' is base-point free. Then, using Proposition 5.1.1, we get

66


In particular, the property that S is small with respect to r does not depend on the choice of symmetric ample line bundles L on A. On An, we can give the NBron-Tate height pairing associated with @ L l p:(L), where pi : An + A is the projection to the i-th factor. For x = (21,. . . , x,) E A(7Qnl (~,x)@y=~p;(L) is given by By abuse of notation, we denote (

, )By=L=,p;(~) by ( ,

)L.

Lemma 7.2.2. Let S c A(R) be a subset that is small with respect to r, and F a finite extension field of K . Let f : A -+ A' be a homomorphism of abelian varieties over R and fn-' : An-' -+ A ~ - ' the homomorphism given by ( X I , . . . ,xn-1) I+ (f ( x i ) , . . . , f (xn-1)). Then, we have the following: (1) The set f (Pn(s, F)) is small; (2) Let bl, . . . , bl be nontorsion points in fn-'(Bn(s, F ) ) ; Then there exists a finite proper subset M such that bi @ f n - l ( P n ( s \ M, F ) ) for any i. Proof: Recall that we have assumed the existence of a finitely generated subof l? with ro in A(K) and ro@ Q = r @ Q. For simplicity, we put group

II . IIL

=

m.+

Let s = y(s) z(s) E S. Since my(s) E ro 2 A(K) for some positive integer m, c(y(s)) - T ( ~ ( s ) )is a torsion point for any c,T E Gal(R/F). Then, J J a ( s )r ( s ) l l ~= llg(z(s)) - r ( z ( s ) ) l l ~ 5 2llz(s)II~.Therefore, we have, for any x E O F ( S ) ~ , ( I , & ( x ) ( (5L 2-llz(s)llL. Let L' be a symmetric ample line bundle on A'. Then, there is a positive constant a with (f (x), f (X))LI5 a(x,x)L. Thus, First, let us see (2). Put p = mini=1,...,l{llbilll'). Then, there is a finite proper subset M of S with P (VS E S \ M ) . II~(S)IIL< 2

Jm

Thus, by (7.2.2.1), we have

Ilf "-'(P~(x>)IILI< P (VX E UqM O F ( S ) ~ ) . @ f n-'(Bn(S \ M, F ) ) for any i.

Hence, bi Next, we consider (1). If fn-'(Dn(S, F)) is infinite, then the assertion of (1) is obvious by (7.2.2.1). Otherwise, let {bl,. . . , bl) be the set of all nontorsion points in fn-'(%(S, F ) ) . Then, by (2), we can find a proper subset M of S with

0 # fn-'(Pn(s\ M 3 ) ) G f n - ' ( P n ( s 1 ~ ) )\ {bl, ... bl). 1

Hence, f n-l (Dn(S,F ) ) contains a torsion point. Thus, f n-l (Pn(S,F))is small.


67

Proposition 7.2.3. Let S be an infinite subset of A ( R ) that is small with respect to r. Then, there exist an infinite subset T of S, a finite extension field F of K , and a positive integer N such that ([N](T),F ) is n-minimized. Proof:

Since A"-'K

is a noetherian space,

{Dn(T, F ) I T is an infinite subset of S and F is a finite extension field of K ) has a minimal element. We fix a pair (T, F ) for which D n ( ~F, ) is minimal. By Theorem 7.2.1, there are abelian subvarieties C1,. . . , CT,torsion points cl, . . . ,c,, and finite nontorsion points bl, . . . , b, such that B,(T, F ) = U:=l (Ci+ci)u{bl, . . . , b,). By Lemma 7.2.2, we can find a finite subset M of T with D,(T \ M , F ) G U;=l(Ci ci). Then by the minimality of (T, F ) , we have

+

T

-

(7.2.3.1)

D n ( T , F ) = U ( c i +ci). i=l

Take a positive integer N with Nc, = 0 for all i. Here, we claim that ([N](T),F ) is n-minimal. Indeed, since T is an infinite set and [N] is a finite morphism, [N](T) is an infinite set. Let TI 5 [N](T)be an infinite subset, and F' be a finite extension field of F. Take T' T with [N](TJ)= TI. Then, we find

Moreover, for any positive integer N', we get

T

=

UG = Bn([N](T),F ) ) .

i= 1 Therefore, ([N](T), F ) is n-minimal.

The following theorem plays a key role in the proof of Theorem 7.1.1 in the next subsection.

Theorem 7.2.4 ([26], [25]). Let S be an infinite subset of A@) that is small with respect to r. Let F be a finite extension field of K . Assume that ( S , F ) is 2minimized. Then, there exists an abelian subvariety C of A( such that Dn(S, F ) = Cn-l for any n 2.

>

Proof:


68


Lemma 7.2.5. Let the notation and the assumption be as in the theorem. Let B an abelian subscheme of AF over F . We assume that there exists a positive integer e with the following properties: For each s 6 S , there exists a subset T(s) of OF(,) x OF(S) such that ,&(T(s)) B(K) and #T(s) 2 #(oF(s) x O ~ ( s ) ) / e .Then, there exists a finite subset M of S and a positive integer N with B ~ ( [ N ] ( S \ MF, ) ) C B ( R ).

c

Proof: Let .rr : A -,A I B be the natural homomorphism. Fix s E S. Take a finite extension field F' of F ( s ) such that F'/F is Galois. Let 4 : G := Gal(F1/F) + OF(S) be the map given by (T H (T(s). Let G,(,) be the stabilizer of ~ ( s by ) the action of G. We set

Then, we have #R = #G,(,) . #G. Moreover, since ( 4 x ~ ) - ' ( T ( s ) ) R, we get #R #(G x G)/e. Thus, [G : G,(,)] 5 e, which means that [F(.rr(s)): F] 5 e. By Lemma 7.2.2, r(D2(S, F)) is small. Then, by Northcott's theorem (TheoF ) ) ) is a finite set. Thus, by Lemma 7.2.2 again, there rem 5.1.4), .rr(B2([~](S, exists a finite subset M & S such that r(D2(S\ M, F ) ) consists of torsion points. Hence, if we take a positive integer N with [ ~ ] ( . r r ( & ( S\ M, F ) ) ) = {0), then we have Dz([N](S\ M , F)) B ( R ) .

>

c

Let us go back to the proof of Theorem 7.2.4. First, let us consider the case n = 2. By the same reason as in (7.2.3.1), we find abelian subvarieties C1,. . . , C, with -

V2 ( S ,F ) = IJ Ci, i=l

because D2(S,F) is stable under any [N]. In order to see e = 1, it is sufficient to find Ci, a positive integer Nl, an infinite subset S1 of S, and a finite extension field Fl of F such that -

F l ) G Ci (17). Let Fl be a finite extension field of F such that every Ci is defined over Fl. For each s E S, put Ti(s) = {x E OFl(s) x OF,(S) ( ,&(z) & Ci(R)). Then, by the pigeonhole principle, there exist i and an infinite subset S' of S such that, for any s E S', #T~(s) 2 #(OFl (s) x OFl (s))/e. Then, by Lemma 7.2.5, there exist an infinite subset S1of S and a positive integer Ni with D 2 ( [ ~ 1 ] ( S lF) ,l ) G c i ( K ) . Then, 52(S,F ) = C. Let us show that - In what follows, we denote Ci by C. Vn(S, F ) = Cn-' for any n 2. Clearly, 2),(S, F) C Cn-'. Thus, it is sufficient to find a positive integer N2, an infinite subset S2of S, and a finite extension field F 2 of F such that v2 ([NlI (S1)

>

-

vn([N2](S2),F 2 ) = cn-'.

69


By Proposition 7.2.3, there exist an infinite subset S 2 of S, a finite extension field F2of F, and a positive integer N2 such that ([N2](S2),F2) is n-minimized. Then, as F2) = Bj. before, there are abelian subvarieties B I , . . . ,BLwith D2([~2](S2), Moreover, since ([N2](S2),F 2 ) is n-minimized, by replacing F 2 by a suitable finite extension field, we may assume that Bi's and C are defined over F2. Now, let us show that D n ( [ ~ 2(Sz), ] F2) = Cn-'. In the same way as before, we can find B j and an infinite subset S' of [N2](S2) such that, for any s E S', there exists a subset U(s) of O F , ( S ) ~with #u(s) 2 #(OF, (s)")/l and Pn(U(s)) ~j ( R ) . In what follows, we denote B j by B. Let ~ ( 4 = ) 0 x . . . x C x . . . x 0 be the q-th factor of cn-l, and put B(Q)= B n c(Q). Since B Cn-I and B is an abelian subcheme, it suffices to see the following claim to conclude the proof.

u:=,

Claim 7.2.5.1. For each q (1 5 q < n - I),

~ ( 4= )

~(4).

Without loss of generality, we may assume q = n - 1. Fix s E S'. For t = (ti, . . . , tn-1) E O F ~ ( S ) ~ - we ' , set Lt(s) = {X E OF(^) I ( t l , . . . tn-1, 2) E U(s)). We choose t' = (ttl,. . . ,ttnPl) E OF, ( s ) ~ -such ' that # ~ t l ( s attains ) the maximum among {#Lt ( ~ ) ) t ~ o ~ ( ~ Then, )n-l. # L ~ ~ ( 's#) o n ( s ) ~ - 2 ' #u(s)

2 #(oF~(s)~)/~,

so that # L ~ , ( s )2 #OF(s)/l. Set T(s) = Ltf(s)x Ltl(s) (x, y) E T(s), we have

c OF(S) x Op(s). Then, for

if )we view B(~-')as a subscheme of A. Since This means that P2(x,y) E B ( ~ - ~ ) ( R P2(T(s)) G B("-')(E) and #T(s) 2 #(OF(S) x O F ( S ) ) / ~ ~ we, get by Lemma 7.2.5 S" of S' and a positive integer N" with that there are an infinite subset ). since D2 ([N"](S1', F2)) = C , we have B("-') = V2([N11](S", F2)) B ( ~ - ~ ) ( TThen, c("-1). 7.3. Proof of Bogomolov plus Lang. We note that, in order to prove Theorem 7.1.1, K can be replaced by any finite extension field of K . Thus, enlarging K , we may assume that X is defined over K and that there exists a subgroup roof r with I?o C A(K) and ro 8 Q = r 8 Q. As in $6.2.1, let G(X) = { a E A ( R ) I a + X = X ) be the stabilizer of X. By comparing A with the quotient A/G(X), we see that it suffices to prove Theorem 7.1.1 when dimG(X) = 0. (For a detailed argument on this comparison, we refer to [25, $31.) And when dimG(X) = 0, what we need to prove is the following.

Claim 7.3.1. If dimG(X) = 0, then X is a finite subset of rdi,.

70

SHU KAWAGUCHI. ATSUSHI MORIWAKI AND KAZUHIKO YAMAKI

Proof: If dim X = 0, then bL(71,.. . ,Y,, x) = 0 for each x X , so that x E rdiv. Thus, it suffices to show dim(X) = 0. In order to lead a contradiction, let us assume ) the contrary, i.e., dim(X) > 0. Then we can find a sequence { s l ) r l in ~ ( 7 7 with the following properties: (a) sl # slt for 1 # 1'; (b) Any infinite subset of {sl I 1 = 1,2,. . .) is Zariski dense in X ; (c) S ~ ( y l , . .,-jq,sl) . 5 111 for all 1. Indeed, by the assumption of Theorem 7.1.1, there exists a sequence {sll)& in X ( z ) with the properties (a) and (c) such that {s'l ( 1 = 1 , 2 , . . .) is Zariski dense in X . Since the number of all subvarieties of XF is countable, a suitable subsequence ) ~ ~ the property (b). of { s ' ~ satisfies By virtue of (c), {sl I 1 = 1,2,. . .) is small with respect to rdiv. Note that for any positive integer N, dimG(X) = 0 if and only if dimG([N](X)) = 0. Then, by Proposition 7.2.3, replacing K by a finite extension field, X by [N](X), and {sl I 1 = 1 , 2 , .. .) by an infinite subset of [N]({sl I 1 = 1 , 2 , .. .I), we get an infinite subset S of x(K) with the following properties: (1) S is small with respect to (2) S is Zariski dense in X ; (3) (S,K ) is 2-minimized.

rdiv;

Then, by Theorem 7.2.4, there exists an abelian subvariety C of AtT such that D,(S, K) = Cn-' for every n 2 2. If dim C = 0, then S A(K). Here we use Lang's conjecture over a number field:

-

If X ( K ) is Zariski dense in X , then X is a translation of an abelian subvariety of A. (For its proof, we refer to [7], [8]. See also Theorem 7.1.2.) Then, X is a translation of an abelian subvariety B of A. Then G ( X ) = B. This contradicts with dim X > 0 and dimG(X) = 0. Next we assume dim(C) > 0. We fix an integer n with n > 2dim(A). Let .rr : A 4 A I C be the natural homomorphism, and put T = .rr(X). Let X$ be the fiber product of X over T in X n . Then, we have the morphism P, : X$ -+ An-' given by ,Bn(xl, . . . , x,) = (x2 - X I , . . . , xn - 2'). Noting that OK(S), C X?, let Y OK(s), in X;. Then, be the Zariski closure of Uses

Therefore, we get dim(XF)

> dim(Y) > dim(cn-').

71


On the other hand, by the assumption that dimG(X) = 0, we have dim(X/T) 5 dim(C) - 1. Thus,

+ dim(T)) - ( n

dim(XF) - dim(cn-l) = (ndim(X/T)

-

1)dim(C)

5 (n(dim(C) - 1) + dim(T)) - ( n - I ) dim(C) = dim(C)

+ dim(T) - n

< 2dim(A) - n

g - 2 , which is a very sharp result in view of Castelnuovo-Severi inequality; cf. Remark 2.11. Therefore one may wonder if there exists a base-point-free pencil of degree d for every d > g - 4 on any double covering of a genus two curve. Putting the problem in a more general setting, we would like to pose the following question.

96

CHAGHO KEEM

Question 3.7 Given a double covering of a curve of genus h, does there exist a base-point-free pencil of degree d f o r every d > g - 2h which is not composed with the involution ? We shall give two partial answers to the above question; one is an example which shows that the statement in Question 3.7 is a little bit too much to expect. On the other hand, we shall exhibit an example of a double covering of a curve of genus 2 on which the statement in Question 3.7 is indeed valid.

Example 3.8 (i) There exists a curve C of genus 7 which is a double covering of a cuwe E of genus 2 with an extra pencil of degree 4 not composed with the given involution. (ii) There also exists a curve C of genus 7 which is a double covering of a cuwe E of genus 2 with no extra pencil of degree 4 other than the one composed with the given involution. Proof. Let C 5 E be a double covering of a curve E with g ( E ) = 2 and g(C) = 7. Let R c E be the branch locus of T . By the Riemann-Hurwitz relation (3.5.2) and base-point-free pencil trick, we have 2 C?E(-gi - p - 4 ) . Let s E H (E,OE(29,1+ 2p + 24)) be a section with the zero locus ( s ) ~= R. Let V be a subspace of H o ( E ,OE(2g; + 2p + 24)) which is the image of the cup product map

s

H o ( E ,OE(gfr + P

+ 4)lB2

H o ( E ,O E ( 2 d + 2P + 24))

and consider a natural morphism

Using the identifications

and

we see that the morphism a induces the map

97

DOUBLE COVERING OF SMOOTH ALGEBRAIC CURVES

with g(1) = s by the algebra structure of R*(OC). We also see that (3.8.2)

I m a = Span(l/,s)eHO(E,OE(gi + p + q ) ) ,

under the identification (3.8.1). Claim 1. Suppose that C is not bi-elliptic and put gi = ~'(9;). Then we have (i) ho(C,Oc(2gi)) 2 4 if and only if S O ~ ( - 2 g i ) , (ii) Suppose S y O ~ ( - 2 g i ) . Then C has at least two complete pencil of degree 4 if and only if s E V. Proof of Claim I. (i) follows directly from the following equality;

hO(C, Oc(2gi)) = hO(C,Oc(.rr*(29i)))= h o w ,.rr*Oc(.rr*(29::))) = ho(E,OE(2gi))

(ii) Since S y O ~ ( - 2 g $ ) ,S

+ ho(E,oE(2gi) 8 s)).

OE(-gi - p - q ) forp+q $ gi and hence the morphism

E + P2 induced by 1g$ + p + q( is birationally very ample. Thus we have an exact sequence

o

-+

Sym'HO(E, OE(gi + p

+ 4 ) ) 4 H O ( E 0,57(2gi , + 2p + 2 q ) ) ,

and we see that dimV = dim Im 6 = dirnIm8 = dim Sym' H o ( E ,OE(g$

+ p + q ) ) = 6.

Therefore it follows that a is surjective if and only if s # V by (3.8.2). Assume that C has a pencil hi # gi and put := IKc - gi - hil. Since C is not bi-elliptic, the morphism 5 : C + n"3 induced by IKc - gi1 = \hi birational onto its image and ( ( C )c @ lies on a quadric surface. By the Riemann-Hurwitz relation IKcl = l.rr*(K~ +S-')l, we have I.rr*(gi + p + q ) I = Ihi k i l . Since

Ici

+ Ici(

+

dimSym'H'(C,.rr*OE(gi + p + q ) ) = ho(C,.rr*O~(2gi + 2 p + 2 q ) = 10, it follows that the map

Sym' H o ( C ,n*OE(gi + p

+ 9))

-

+ + 2q))

Ho(C,. r r * o ~ ( 2 g i 2p

is not surjective. Hence the map a is not surjective and we have s E V. Conversely, assume that s E V. Then the morphism a is not surjective and hence the image of the morphism C -+ P3 induced by the birationally very ample I./r*(gi+p+q)I is contained in a quadric surface S c P3. In case S is a cone, there is a pencil hi such that ho(C,Oc(2hi)) = 4. From the assumption S O ~ ( - 2 g i ) ,it

98

CHAGHO KEEM

follows that ho(C,O c ( 2 g i ) ) = 3 by (i) and hence gi # hi. In case S is a non-singular quadric, we also have two pencils of degree 4 corresponding to the rulings of S and this finishes the proof of Claim 1. Note that a curve C of genus g 5 7 which is a double covering of a curve of genus 2 may be also bi-elliptic, whereas a curve of genus g 2 8 cannot be both bi-elliptic and a double covering of a curve of genus 2 by Castelnuovo-Severi inequality. The following Claim 2 provides a simple criteria for a double covering of a curve of genus 2 being bi-elliptic. Claim 2. Let C be a curve of genus g = 7 which is a double covering of a curve of genus 2 with an involution L induced by the covering. If C is also bi-elliptic, then the bi-elliptic involution r commutes with L; i.e., LT = TL. Proof of Claim 2. Let T I : C 4 El be a double covering, where El is an elliptic curve and let r be the involution induced by "1. Consider the double covering ~2 : C -+ E 2 induced by L - ~ T L . We remark that Q E C is invariant under L - ~ T L if and only if L(Q)is invariant under r. Thus, if R is the ramification locus of T I then L(R)is the ramification locus of 7r2. It follows that L - ~ T L is also a bi-elliptic involution by the Riemann-Hurwitz formula. By Castelnuovo-Severi inequality, bi-elliptic involution of C is unique, whence LT = TL and we are done with the proof of Claim 2. By the Claim 2, it follows easily that if a double covering C 4E is also bi-elliptic then there is an automorphism on E which lifts to the bi-elliptic involution r via T . Finally, we take a curve E of genus 2 such that Aut(E) = {o,l~} where a is the hyperelliptic involution on E. Let

c = Spec(OE 69 O E ( - g ;

+

-

p -4))

with p q # gf where p , q E E are not fixed points of u. If C is bi-elliptic with a bi-elliptic involution r , then a lifts to 7; note that 1~ does not lift to r via n. Then it follows that CT*OE(-gi - p - q ) OE(-gi - p - q ) , which implies a ( p ) + u ( q ) = p + q and hence a ( p ) = p , a(q) = q contrary to the choice of p , q E E as non-fixed points of a. Therefore C cannot be bi-elliptic. ~ reduced and (i) Choose t l , tz E Ho(E,O E ( g f p 4 ) ) such that ( t l ) ~( ,t z ) are (t1)o n ( t z ) o = 8. Then R = ( s ) ~ where , s = t l t 2 , is also reduced. We note that s E V . Therefore the curve C = Spec(OE@ OE(-9; - p - 4 ) ) with the branch locus R is non-singular, which has at least two pencils of degree 4 by Claim 1. (ii) Recall that V Ho(E,O E ( 2 9 ; + 2 p + 2 q ) ) and take s # V such that R = ( s ) ~ is reduced. Then the curve C with branch locus R has no extra pencil of degree 4 other than T*(g;) by Claim 1. W It should be remarked that Example 3.8 has been constructed and used for another purpose in a joint work with Dr. K. Cho and Professor Akira Ohbuchi; cf.

+ +


99

[CK02].

Example 3.9 Given a curve E of genus h 2 1 and an integer g 2 8h, there exists curve C of genus g which is a double covering of E possessing a base-point-free pencil g i not posed with the involution for every d 2 g - 2h 1.

+

Proof. On a ruled surface S := E x B1 we consider a linear system

where D E Eg-2h+l and po E B'. Since g >> h, (r(is very ample. Therefore there exists a smooth irreducible curve C E (r(by a theorem of Bertini. Using adjunction formula, one computes easily that pa(C) = g. Denoting 7ri the restrictions to C c S of the projection morphisms from S onto its factors, we see that deg7rl = 2 and deg 7r2 = g - 2h 1. It is now clear that the morphism 7r1 is a double covering of E and "2 induces a base-point-free pencil of degree g - 2h 1 which is not composed with the involution. Thus we are done ford = g - 2 h f l . For the cased > g - 2 h + l 1 we leave as an exercise t o the interested reader. [Hint; use a method used in the proof of Proposition 5.1, or use Lemma 5.21

+

+

4 Double covering of a general curve of higher genus. In previous sections, we mainly dealt with double coverings of a curve genus h _< 2. For a double covering of a curve of genus h 2 3, the situation becomes quite complicated and we will only present results which are relatively weaker. The following theorem may be considered as a generalization of our results in previous sections except the fact that the base curve of the double covering being considered in this section is a general curve of genus h.

Theorem 4.1 (Ballico-Keem, [BKl, Theorem 0.11) Let C be a smooth algebraic curve of genus g which admits a two sheeted covering onto a general curve E of genus h, g 2 5h 1. Then there exasts a base-point-free pencil of any degree d 2 g - h which is not composed with the given involution.

+

A proof of Theorem 4.1 requires several preparatory results and we begin with the following theorem due t o Matelski [Mat] ; see also [Ho2, Corollary 11.

Lemma 4.2 Let C be a smooth algebraic curve of genus g 2 4j + 3 , j 2 0. If dimWj(C) = d - 2 - j for some d such that j + 2 5 d 5 g-1-j, then dimW&+2(C)= j.

100

CHAGHO KEEM

Lemma 4.3 Let C be a curve of genus g which is a double covering of a general curve E of genus h. If g >_ 4 h then WiVh(C)has the expected dimension p ( g h, 9 , 1) = g - 2h - 2. Proof. Suppose Wi-h(C) does not have the expected (and minimal) dimension g - 2h - 2. Set dimW,'_,(C) = ( g - h ) - 2 - j 2 g

(4.3.1)

2h - 1

-

Then by substituting d = g - h in Lemma 4.2, we easily see that the numerical hypothesis of Lemma 4.2 are satisfied. Thus by Lemma 4.2, we have dimW&+2(C)= j and hence dimWdh(C) > _ 2 h - ( 2 j + 2 ) + j = 2 h - j - 2 . On the other hand, by Castelnuovo-Severi inequality and by the hypothesis g 2 4 h , every component of Wih(C) is of the form r*(CA(E)) W Z ~ - ~ ~for ( Csome ) n, where 7r : C -+ E is the double covering and CA(E) is a component of WA(E). But dimr*(CA(E)) W2h-2n(C) = p ( n , h, 1 ) 2h - 2 n = h - 2

+

+

+

since the base curve E is general. Consequently we have dimW&(C) = h - 2 2 2h - j

-

2

which is contradictory to (4.3.1). H

Lemma 4.4 Let C be a curve of genus g which is a double covering of a general curve E of genus h. Let g 2 5 h f l . Then the components of WiPh(C)whose general elements have base points are of the form r*(CAlz(E)) Wg-h-n(C) with CAIz(E) a component of W$2(E), where n is even and 2[y] _ n - h - 2

101


where po : Ho(C,L ) 8 Ho(C,KcL-') Ho(C,K c ) is the usual cup product map. By the base-point-free pencil trick, we have --f

dim(Impo)' = g - dim(Imp0) = g - ho(C,L)hl(C,L )

+ dim(Kerp0)

+ 1)+ ho(C,K c L - ~ )= ho(C,L 2 )- 3 2 n - h - 2. Hence ho(C,L 2 ) 2 n - h + 1 which implies W,",-'(C) 2 n - h - 2. By reducing =g

-

2(g - n

to pencils we have 1 dimWA+h+l(C) = d i ~ n W ~~-(~-h-~ 2)72( C - )h - 2

Note that n 5 g - h - 1, thus n

(1)If n

+ ( n- h - 1) = 2(n - h) - 3.

+ h + 1 5 g. We consider the following two cases:

+ h + 1 = g , then by passing to residual series

dimW;+,+,(C)

= dimWg-2(C) = g - 2 2 2(n - h) - 3

g

5 4h + 3,

contradictory to the genus bound g 2 5h + 1. ( 2 ) If n h 1 5 g - 1, we have

+ +

+ + 1) - 2 - 1e n 5 3h + I by H.Martens' well-known theorem; Theorem 1.5. Thus n 5 3h + 1 5 g - 2 h by 2 ( n - h ) - 3 5 dimW,'+h+l(C) 5 ( n h

+

the genus bound g 2 5h 1, and again by the Castelnuovo-Severi inequality every element of ,Ei is a pull-back of a gAI2 on E , i.e. EL = r*(E:/,(E)), where CA12(E) is a component of W,'12(E). Since dimxi = dimr*(,EAl2(E)) = dim(C;,2(E)) = dim(W&,(E)) = n - h - 2 5 h, we have n 5 2h

+ 2 , and [y] 5 f since E is general.

Proof of Theorem 4.1. We use the same notations and conventions explained in the beginning of Section 2; u : c d -+ J ( C ) is the abelian sum map, 8 is the class of the theta divisor in J ( C ) ,u* : H * ( J ( C ) , Q )-+ H*(Cg-h,()) is the homomorphism induced by u. Also recall that by fixing a point p on C , one has the map L : Cd-1 -+ c d defined by L ( D )= D + p . We denote the class of L ( c d - 1 ) in c d by x. Let r : C -+ E be the double covering. By Lemma 4.4, the only component of c i - h (C Cg-h) consisting of divisors whose complete linear series have base points are the components of the equi-dimensional locus T * ( E ~ , ~Cg-h-n, )

+

102

CHAGHO KEEM

[y]

+

5 n/2 5 h 1, n = even. We denote an and Lin by the class of .rr*(E’ ) + Cg-h-n in Cg-h and the class of T*(EA/~) in cnrespectively. Because 4 2 Cg-h is of pure and expected dimension p(g - h, g, 1) 1, the class of Cj-h is well-known (cf. [ACGH, page 326]), namely with

+

(4.1.1)

As in Section 2, let

be maps induced by the assignments

B k ( 2 ) := { E E c d - k : D - E

2 0 for some D

E 2)

given a cycle 2 in c d . Thus by the Formula 2.1 and Formula 2.4 (Push-pull formulae for symmetric products), we have

and

= (6,. Bg-h-n(Z

g-2h-1 g-h-n

g-2h-1

by noting the fact that (6, . xn-h-l )c, = 1 On the other hand, by (4.1.1) 1 (xg-h

Zg-2h-1 ’

)cg-h

1 ((h!oh+l- ( h - h!(h l)!

+

-

9! 1 ( 9 - h - I)! ( h + l)!

+ g! (9-h)!

. Zg-2h-1 1

h!

1Cg-h

103


where the last equality comes from the fact that (zd-P is a consequence of the Poincari: formula; Formula 2.3. Finally a messy calculation yields

. @)c,= g ! / ( g - p)!,which

g! , _ _1_ _ _ _ _9!. _ (g-h-l)! ( h f l ) ! (9-h)!

g-2h-1

I h!

n.=euen

and this shows that there exists a component in cj-h other than those of the form

+

r * ( E 1 ) Cg-h-n which in turn proves that there exists a divisor of degree g - h 4 2 on C which moves in a basepoint-free pencil and whose complete linear series is not composed with the given double covering. By the excess linear series result [ACGH, page 3291, one has dimW&,(C) 2 dimW,T(C) - r - 1. Because dimW;-,(C)

+ 1, we have I dimWjP1(C)+ 2 L p ( d - 1,, g , 1) + 2 = p ( d , g , 1).

= p(g

p ( d , g , 1) I dimW;(C)

- h, g , l),for any d 1 g

-h

Thus dimWi(C) = p ( d , g , 1) for any d 2 g - h, which implies that there exists a complete base-point-free pencil of degree d which is not composed with the double covering.

5

Irreducibility of W,'(C) for double coverings.

In Theorem 4.1, we showed not only the existence of a base-point-free pencil of degree d 2 g - h which is not composed with the given involution but also the reducibility of W;-h(c) for a double covering of a general curve of genus h. Therefore one may wonder what would be the range of d in terms of g and h for the irreducibility of W i ( C ) . It turns out that Theorem 4.1 is quite a sharp result; i.e. W i ( C )is irreducible for every d 2 g - h 1 for a double covering of a general curve of genus h. We first prove the following proposition, which is an intermediate step toward the proof of the irreducibility of W,'(C) in the range d 2 g - h 1 for a double covering of a general curve of genus h.

+

+

Proposition 5.1 (Ballico-Keem, [BK2, Proposition 1.51) Let C be a n irreducible smooth algebraic curve of genus g which admits a t w o sheeted covering r 1 C -+ E o n t o a general curve E of genus h > 0, g 2 5 h - 2. T h e n the variety W i ( C ) of pencils of degree d 2 g - h 1 o n C i s generically reduced and a general element of a n y component of W j ( C )i s base-point-free.

+

104

CHAGHO KEEM

We need the following lemma due to Coppens which is an application of the so-called excess linear series argument.

Lemma 5.2 (M. Coppens [C, Theorem 41) Let C be an algebraic curve of genus g. Suppose that Wl(C) has the expected dimension, i.e. dimWd(C) = p(d,g,r) := g - ( r + l ) ( g - d + r ) . ThendimWi+T1(C) = p ( d + l , g , r ) and Wl+l(C) is irreducible (resp. reduced) if Wl(C) i s irreducible (resp. reduced). The following is a well-known criteria for the irreducibility of W,'(C) which follows from [FL, Remark 1.81.

Lemma 5.3 Let C be a smooth algebraic curve of genus g. Suppose that W$(C) has the expected dimension p(d,g, r ) > 0 and that the codimension of the singular locus Sing W,'(C) is at least two. Then W,'(C) is irreducible. We also need the following dimension theoretic statement for Wd; [CKMl, Theorem 3.3.11.

Lemma 5.4 Let C be a smooth algebraic curue of genus g . Let n E N, g 2 2(n+1)2 anddimW;+,(C) < l . T h e n d i m W d ( C ) I 2 d - 6 - g f o r g - n < d L g . Proof of Proposition 5.1. By Lemma 5.2, it is enough to prove Proposition 5.1 for d = g - h 1. Let C be a component of Wj-h+l(C) whose general element has a base point. Set C = Ck+Wg-h+l-n(C) for some n 5 g - h, where E i is a subvariety of Wi (C) whose general element is base-point-free. Hence

+

dimEA = dimC - ( g - h + 1 - n) = n - h - 1 Let L E be general. Then dimTLCA base-point-free pencil trick, dirn(Imp0)' = g - ho(C,L)hl(C,L ) = g - 2(g - n

2 dimCk

>

n - h - 1, and hence by the

+ d'im Ker 110

+ 1)+ ho(C,K L - 2 ) = ho(C,L2) - 3 2 n - h - 1.

Then ho(C,L 2 ) = n-h+2 for general L E C i and hence dim WFLh+'(C)2 n-h-1. By taking off ( n - h ) general points on C , we have (5.1.1) Note that n

dimW;+h(C) 2 2(n - h ) - 1.

+ h 5 g and we distinguish the following two cases.


(i) If n

105

+h =g, dimWj(C) = dimWg-2(C) = g - 2 2 2(n - h) - 1 = 2(g - 2h) - 1,

which is contradictory to the genus bound g 2 5 h - 2. (ii) If n h 5 g - 1, 2 ( n - h) - 1 5 dim Wi+,(C) 5 n h - 3 by (5.1.1) and by H. Martens theorem. Then by the genus bound g 2 5h - 2, n 5 3 h - 2 5 g - 2h and hence by Castelnuovo-Severi inequality, one has ,XA c .*(WA(E)). On the other

+

+

hand dim W i ( E ) = n - h - 2 since E is general. Hence

2

n - h - 1 = dimCA 5 dim.rr*(WA(E))= n - h - 2 which is a contradiction. And this proves that a general element of any component of W j ( C )is base-point-free. For the generically reducedness of Wj-h+l(C), we only need to compute the dimension of the Zariski tangent space TLW;-,+~(C) at a general L. Suppose 1 dimTLWg-h+l(C) = dim(Imp,)'

> dimW;-h+l(C)

= g - 2h

for a general L E Wj-h+l(C). L being base-point-free, it easily follows that h0(C,KL-') 2 1 for general L E Wj-h+l(C) by the base-point-free pencil trick. Then we have

g - 2h 5 dimWj-h+l(C) 5 dimW'h-4(C) = 2h - 4 , which is contradictory to the genus bound g 2 5h - 2. 1 We now ready to prove the main theorem in this section.

Theorem 5.5 (Ballica-Keem, [BK2, Theorem 1.11) Let C be a smooth algebraic curve of genus g which admits a two sheeted covering T : C .+ E onto a general curve E of genus h > 0 , g 1 max{2h2, 5 h 3 ) =: E ( h ) . Then the variety W;(C) of pencils of degree d on C is generically reduced and irreducible with the expected dimension for all d 2 g - h 1.

+

+

Proof. We first claim that W;-h+l(C) is equi-dimensional of the expected dimension p ( g - h 1, g , 1 ) = g - 2h. Indeed, in Lemma 4.3, it is proved that Wj-,(C) has the expected dimension if g 14h. Hence the same is true for Wi-h+l(C) by Lemma 5.2. Therefore, by 5.2 it is sufficient to prove the theorem only for W;-h+l(C).

+

106

CHAGHOKEEM

By a result of Mayer [MI, one has sin!z(W-h+l(CN I)W,-h+l(C). We now claim that dim Wl-h+l(C) I p(g - h

+ 1, g , 1 )

-

2 =g

-

2h - 2

+

Suppose h = 2e 1 is odd and take n = h - 1 = 2e. Then by the Castelnuovo-Severi inequality, one has

WA+3(C) = Wie+3(C) = ..*W,+,(E)

+

+ WI(C).

Because E is general, dimW,+l(E) = p ( e 1,h, 1 ) = -1. Therefore W;+,(C) = 0 and hence dimW;+3(C) < 1. By taking d = g - h 2 in Lemma 5.4, one has

+

dimWi-h+l(C) 5 dimW&+z(C) 5 2 ( g - h

+ 2 ) - 6 - g = g - 2 h - 2.

Suppose h = 2e and take n = h - 1. Again by Castelnuovo-Severi inequality, one has ~:+3(c) = Wie+z(C) = ..*w.+~(E). Since E is general dim W;+,(E) = p ( e + l , h, 1) = 0 and hence dim W;+,(C) = 0 < 1. By taking d = g - h + 2 in Lemma 5.4, one also has dimWi-h+l(C) 5 dimWg-h+z(C) 5 2 ( g - h

+2) -6 -g =g - 2h - 2

and this finishes the proof of the claim. We now suppose that the singular locus Sing Wi-h+l(C) has codimension at most one in Wj-h+l(C).By the above claim we may also assume that Sing W,-h+l ( C ) I)W,-h+l (C)u A, where A is an irreducible closed subvariety of W,-,+, (C)such that dimA 2 g-2h-1 and A Wj-h+l(C). We break up the proof into the following two cases. (i) Assume that a general element of A has no base point, and choose L E A a general element. Since L E SingWSph+,(C) and by the base-point-free pencil trick, one has dimTLW~-h+l(C)= dim(Imp0)' = g - 2 h =g

-

2h

+ ho(C,KL-')

+ dimKerp0 > dimWi-h+l(C)

=g

-

2h

107


where po : Ho(C,L ) 8 Ho(C,KL-') + Ho(C,K ) is the usual cup-product map; this follows from a general theory of special linear series (cf. [ACGH, Prop.(4.2), page 1891 ). Therefore h 0 ( C , K L p 2 > ) 0 and hence KL-' E W2hP4(C) for general L E A. We then have g - 2h - 1 I dimA I dim W2h-4(C) = 2h - 4,

which is contradictory to the genus bound g 2 E(h). (ii) Assume that A c Wi-h(C) W1, i.e. a general element of A has a base point. Note that dimA = g - 2h - 1 since W,l_h+l(C) is generically reduced by Proposition 5.1. Because dim W;-h(C) = g - 2h - 2 , there exists a component Y of WjPh(C)such that A = Y Wl(C), dimY = g - 2h - 2. (ii-a) Suppose Y is not of the form Y' W l ( C )for some Y' c Wi-h-l(C). Then a general M E Y is base-point-free and M 8 O ( p ) ,p E C general, has only one base point p . By the base-point-free pencil trick applied to the cup-product map

+

+

+

PO : Ho(C,M 8 U ( p ) )8 H o ( C ,KM-' 8 U ( - p ) )

+

Ho(C,K ) ,

Kerpo No(C,KM-' 8 U ( - p ) ) # 0 since M 8 O ( p ) E A c SingWil_h+l(C). Therefore we have KM-' 8 U ( - p ) E W2h-3(C) for general M E Y and p E C. From this we get an inequality,

g

-

2h - 1 = dimA 5 dirnW2hp3(C)

+ 1 = 2h - 2 ,

which is contradictory to the assumption that g 2 E(h). (ii-b) Suppose Y is of the form Y'+W1 (C) for some Y' c Wiph-' (C). We claim that Y is of the form a*(CA12(E)) + Wg-h-n(C) with Ci12(E)a component of W:12(E),

+ 2. Proof of Claim. Because Y is of the form Y'

where n is even and 2[y] 5 n 5 2h

+ W l ( C )for some Y' c Wi-h-l(C),

Y is a component of Wj-h(C)whose general element has a base point. Then Y = CA Wg-h-n(C) for some n, 0 I n 5 g - h - 1, where CA is a subvariety of W;(C) and a general element of Ck is base-point-free. We will first argue that n is relatively small compared to g. Because Y has dimension g -2h -2, one has dim EA = n -h -2, otherwise

+

g-2h-2

= d i m Y =dim(Ck+Wg-h-n(C))

# (n-h-2)+(g-h-n)

=g-2h-2.

Let L be a general element of CA. By the standard description of the Zariski tangent space to the variety W i , we have dim(Imp0)' = dimTL(Ck) 2 dimCk 2 n - h

-

2

108

CHAGHO KEEM

where po : Ho(C,L ) @ H o ( C K , L - l ) + H o ( C ,K ) is the usual cup-product map. By the base-point-free pencil trick, we have dim(Imp0)' = g - dim(Imp0) = g - ho(C,L)h'(C,L) =g

-

2(g - n

Hence ho(C,L 2 ) 2 n - h pencils we have

+ dim(Kerp0)

+ 1)+ ho(C,K L P 2 )= ho(C,L 2 ) - 3 2 n

-

h - 2.

+ 1 which implies W;Lh(C) 2 n - h - 2. By reducing to

dimWA+h+l(C) = dimWin-(n-h-l)(C) 2 n - h - 2

+ ( n- h - 1) = 2 ( n - h) - 3.

+ +

Note that n 5 g - h - 1, thus n h 1 5 g . We consider the following two cases: (1) If n h 1 = g , then by passing to residual series

+ +

dimWA+h+l(C) = dimWg-2(C) = g - 2 2 2(n - h ) - 3

g 5 4h+ 3,

contradictory to the genus bound g 2 E ( h ) . ( 2 ) If n h 1 5 g - 1, we have

+ +

+ + 1) - 2 - 1 H n 5 3h + 1 by H.Martens' theorem; cf. Theorem 1.5. Thus n 5 3h + 1 5 g - 2 h by the genus 2 ( n - h ) - 3 5 dimWA+h+l(C)

(n h

bound g 2 ~ ( h and ) , again by the Castelnuovo-Severi inequality every element of CA is a pull-back of a gA/2 on E , i.e. CA = 7r*(C;/,(E)), where CkI2(E) is a component of w $ ~ (E ) . Since ) dim(C&2(E))= dirn(W,'/,(E)) dimCi = d i m ~ * ( C : / ~ ( E )=

+

we have n 5 2 h 2 , and claim. We next claim that

=n-h -2

5 h,

[y] L 3 since E is general. This finishes the proof of the

hO(C,(7r*N)@'2) = hO(E,N @ ) = n - h + 1

+

for N E Ck12(E)general, 2 [ y ] 5 n 5 2 h 2: Since E is general, WLj2(E) is , ho(E, N g 2 ) = n - h 1. Suppose reduced at a general point N E C k / 2 ( E ) hence that ho(C,( T * N ) @>~ho(E, ) N@'), i.e. 7r*Ho(E,N@') 2 Ho(C,(7r*N)@').Then it follows that the complete linear system (7r*N)@"is not composed with 7r. Thus C has a base-point-free g i which is not composed with 7r, 2

+

5 deg(7r*N)@2- ( n - h ) = 2 n - ( n- h) = n + h 5 3h + 2


109

by subtracting n - h generically chosen points on C. But this is contradictory to the Castelnuovo-Severi inequality since g 2 E ( h). Now consider a general M E Y = 7r*(CA/,(E)) Wg-h-n(C),and M 8 O ( p ) E A = Y + W l ( C ) , p E C general. Then

+

M = 7r*N 8 O(Pl+ . . ' + Pg-h+), M 8 O ( p ) = 7r*N 8 O(p1

+ . . . + Pg-hpn + p ) ,

where N E EA/,(E). Applying the base-point-free pencil trick to the cup-product map PO: Ho(C,

8 O(p))8 Ho(C,KIM-' 8 O ( - p ) ) + Ho(C,K ) , one has

Kerpo ~ : ' o ( C , K 8 ( 7 r * N ) ~ - 2 8 0 ( - p - .l . . P ~ - ~ -- ~p ) ) . On the other hand, from the previous claim ho(C,(7r*N)'@)= n - h + 1 and hence

ho(C,K 8 (r*N)@'-')= g Since p i , . ' ,Pg-h-n, p E

-

n - h.

c have been chosen generically, we have

- ~O(-pl - . . . pgPhpn - p ) ) = 0. dim Ker po = ho(C,K 8 ( X * N ) @8 But this is contradictory to the fact that M 8 O(p) E Sing Wj-h+l(C),i.e. dimKerp0 > 0. So far, we have shown that Sing Wil_h+l(C) has codimension at least two in W,&h+l(C).By [FL, Remark 1.81, we finally conclude that Wjl_h+l(C) is irreducible.

6

Epilogue; an odd end with general k-gonal curves.

In previous sections we mainly played with double coverings and saw that not only the Liiroth's semigroup S ( C ) but also the invariant d o ( C ) may depend on the genus of the base curve. Therefore one may wonder what it would be like if we consider a general k-gonal curve instead of a double covering. Fortunately, we already have sufficient knowledge to answer this question.

Theorem 6.1 (Arbarrello-Cornalba [AC, Theorem 2.61) Let C be a general k-gonal curve of genus g , and k < [(g + 3)/2], i.e. p(g, 1,k) < 0. T h e n C has a a unique g k , and each base-point-free (incomplete) pencil g: o n C with p ( g , 1 , d ) < 0 is compounded of the 9;. Furthermore, wj(c)= wi(C)+ W d - k for k 5 d and p(g, 1,d ) < 0. I n particular W j ( C )is irreducible of dimension d - k if p(g, 1, d ) < 0.

110

CHAGHO KEEM

According to Theorem 6.1, any complete and base-point-free pencil on a general k-gonal curve C different from the g i must have non-negative Brill-Noether. number. However Theorem 6.1 does not necessarily guarantee the existence of such pencils. The following Brill-Noether type theorem provides the existence of complete and base-point-free pencils.

Theorem 6.2 (Coppens-Keem-G.Martens [CKMZ, Theorem 2.21) Let C be a general k-gonal curve of genus g , 3 5 k < [ ( g 3)/2], and let d 5 g be an integer such that the Brill-Noether number p(g, 1,d ) = 2(d - 1) - g is non-negative; i.e. n 2 ( g / 2 ) 1. Then

+

+

and each irreducible component Z of Wi(c)different from W i ( C )+ w&k(c)has the minimal possible dimension dim2 = p ( g , 1,d ) . Theorem 6.2 easily yields the following corollary.

+

Corollary 6.3 Let C be a general k-gonal curve of genus g , 3 5 k < [ ( g 3)/2]. Then for any d E M with ( g / 2 ) 1 5 d 5 g there is a complete and base-point-free pencil g: on C such that 29: is non-special.

+

Corollary 6.3 together with Theorem 6.1 easily yields the following result which completely determines the invariant do(C) for a general k-gonal curve.

Proposition 6.4 Let C be a general k-gonal curve of genus g . Assume that 3 5 k < [ ( g 3)/2]. If is a multiple of k , then do(C) = [%I. If is not a multiple of k , then do(C) =

+

[q] [q].

[q]

REFERNCES [A]

R. Accola, Topics in the theory of Riemann surfaces, Lecture Notes in Mathematics, 1595 (1994), Springer Verlag.

[AC] Arbarello, E and Cornalba, M., Footnotes to a paper of B. Segre, Math. Ann. 256 (1981), 341-362. [ACGH] Arbarello, E., Cornalba, M., Griffiths P. A. and Harris J., Geometry of Algebraic Curves I, Springer Verlag, 1985. [BKl] E. Ballico and C. Keem, O n multiple cowering of irrational curves, Arch. Math. 65 (1995), 151-160. [BK2]

E.Ballico and C. Keem, Variety of linear systems on double covering curves, Journal of Pure and Applied Algebra. 128 (1998), 213-224.

DOUBLE COVERING OF SMOOTH ALGEBRAIC CURVES [C]

111

Coppens, M., Some remarks on the schemes W l , Annali di Matematica pura ed applicata (IV)157 (1990), 183-197.

[CKMl] Coppens, M., Keem, C. and Martens, G., Primitive linear series o n curves, Manuscripta Math. 77 (1992), 237-264. [CKMZ] Coppens, M., Keem, C. and Martens, G., Primitive length of a general k-gonal curve, Indag. Mathem., N.S. 5(2) (1994), 145-159. (CKOl] Cho, K., Keem C. and Ohbuchi A,, O n the variety special linear systems of degree g - 1 on smooth algebraic curves, International Journal of Mathematics, to appear.

[CK02] Cho, K., Keem C. and Ohbuchi A,, Variety of nets of degree g - 1 on smooth curves of low genus, Journal of the Mathematical Society of Japan, to appear. [FL] Fulton W. and Lazarsfeld R., O n the connectedness of degeneracy loci and special divisors, Acta Math. 146 (1981), 271-283 [GH] Griffiths, P. and Harris, J., The dimension of the variety of special linear systems on a general curve, Duke Math. J . 47 (1980), 233-272. [Hol] R. Horiuchi, O n the existence of meromorphic functions with certain low order on nonhyperelliptic Riemann surfaces, J . Math. Kyoto Univ. 21-2 (1981), 397-416. [Ho2] Horiuchi, R. Gap orders of meromorphic functions on Riemann surfaces, J . reine angew. Math. 336 (1982), 213-220. [K]

Keem, C., O n the variety of special linear systems o n a n algebraic curve, Math. Annalen 288 (1990), 309-322.

[KL] Kleiman, S. and Laksov, D., O n the existence of special divisors, Am. J. Math. 94 (1972), 431-436. [MI Mayer, A.L., Special divisors and the Jacobian variety, Math. Ann. 153 (1964), 163-167. [Mat] Matelski J.P., On geometry of algebraic curves, Ph.D. Thesis, Princeton (1978). [MH] Martens H., O n the varieties of special divisors on a curue, J . Reine Angew.Math. 233 (1967), 111-120.

[PI

Park, S., Special linear series on 5-gond curves and bi-elliptic curves, and very ample series on general k-gonal curves, Ph.D. Thesis, Seoul National University (2002).

[Sh]

V. Shokurov, Distinguishing Prymians from Jacobians, Invent. math. 65 (1981), 209-219.

Department of Mathematics Seoul National University Seoul 151-742 South Korea email:[email protected]

ALGEBRAIC SURFACES WITH QUOTIENT SINGULARITIES - INCLUDING SOME DISCUSSION ON AUTOMORPHISMS AND FUNDAMENTAL GROUPS

JONGHAEKEUMAND DE-QI ZHANG

INTRODUCTION We work over the complex numbers field C. In the present survey, we report some recent progress on the study of varieties with mild singularities like log terminal singularities (which are just quotient singularities in the case of dimension 2; see [KMM]).Singularities appear naturally in many ways. The minimal model program developed by Mori et a1 shows that a minimal model will inevitably have some terminal singularities [KMo]. Also the degenerate fibres of a family of varieties will have some singularities. We first follow Iitaka's strategy to divide (singular) varieties Y according to the logarithmic Kodaira dimension .(Yo) of the smooth locus Y oof Y . One key result (2.3) says that for a relatively minimal log terminal surface Y we have either nef K y or dominance of Y oby an &ne-ruled surface. It is conjectured to be true for any dimension [KMc]. In smooth projective surfaces of general type case, we have Miyaoka-Yau inequality cf 5 3C2 and Noether inequalities: p , 5 (l/2)cf 2, cf 2 (1/5)c2 - (36/5). Similar inequalities are given for Y oin Section 4; these will give effective restriction on the region where non-complete algebraic surfaces of general type exist. In Kodaira dimension zero case, an interesting conjecture (3.12) (which is certainly true when Y is smooth projective by the classification theory) claims that for a relatively minimal and log terminal surface Y of Kodaira dimension ,(Yo) = 0, one has either r l ( Y o )finite, or an etale cover Zo + Y owhere Zo is the complement of a finite set in an abelian surface 2. Some partial answers to (3.12) are given in Section 3. The topology of Y o is also very interesting. We still do not know whether r1 of the complement of a plane curve is always residually finite or not. Conjecture (2.4) proposed in [Z7] claims that the smooth locus of a log terminal Fano variety has finite topological fundamental group. This is confirmed when the dimension

+

113

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JONGHAE KEUM AND DE-$1 ZHANG

is two and now there are three proofs: [GZl, 21 (using Lefschetz hyperplane section theorem and Van Kampen theorem), [KMc] (via rational connectivity), [FKL] (geometric). The other interesting topic covered is the automorphism groups. Recent progress in K3 surface case is treated in Section 5 . For generic rational surface X of degree 5 5, it is classically known that IAut(X)l divides 5!. However, when Y is a log terminal del Pezzo singular surface of Picard number 1, it is very often that Aut(Y) contains Z/(p) for all prime p 2 5 (see [Z9] or (6.2)).

TERMINOLOGY AND NOTATION (1). For a variety (2). A (-n)-curve (3). For a divisor Supp D. (4). For a variety

V we denote by Vo = V-Sing V the smooth locus. C on a smooth surface is a smooth rational curve with C2 = -n. D , we denote by # D the number of irreducible components of V, the e(V) is the Euler number. SECTION1. PRELIMINARIES

(1.1). Let Vo be a nonsingular variety and let V be a smooth completion of Vo, i.e., V is nonsingular projective and D := V \ Vo is a divisor with simple normal crossings. If H o ( V , m ( K v D ) ) = 0 for all m 2 1, we define the Kodaira (logarithmic) dimension 6(V0) = -m. Otherwise, Im(Kv D)l gives rise to a rational map q m for some m and we define the Kodaira dimension n(Vo) as the maximum of dim( q m (V')) . The Kodaira dimension of Vo does not depend on the choice of the completion V [I3, $11.21. Also 6(V0) takes value in {-m, O , l , . . . ,dimVo}. Vo is of general type if .(V0) = dimVo.

+

+

+

p,(Vo) = ho(V,Kv D ) is called the logarithmic geometric genus which does not depend on the choice of the completion V [ibid.]. (1.2). (a) Let G GL2(C)be a non-trivial finite group with no reflection elements. Then C 2 / Ghas a unique singularity at 0 (the image of the origin of the f i n e plane C 2 ) . A singularity Q of a normal surface Y is a quotient singularity if locally the germ (V, Q) is analytically isomorphic to ( C 2 / G D , ) for some G. Quotient singularities are classified in [Br, Satz 2.111. (b) When dimY = 2, the Q in Y is a quotient singularity if and only if it is a log terminal singularity [Ka2, Cor 1.91.

ALGEBRAIC SURFACES WITH QUOTIENT SINGULARITIES

115

Q is a Du Val (or rational double, or Dynkin type A D E , or canonical, or rational Gorenstein in other notation) singularity if G SL2(C) (see [Du], [Rel]). In (1.3) - (1.6) below, we assume that Y is a normal projective surface with at worst quotient singularities.

(1.3). Let f : ? Y be the minimal resolution and D the exceptional divisor. We can write f * K y = K p D* where D' is an effective Q-divisor with support in D . Write D = Di with irreducible Di and D* = diDi. ---f

x:=l

+

cZl

Lemma. (1) Each Di is a (-ni)-curve for some ni 2 2. (2) 0 5 di < 1. (3) di = 0 holds if and only if the connected component of D containing Di is contracted to a Du Val singularity on Y (i.e., f ( D i ) is a Du Val singularity). Proof. (2) follows from the fact that a quotient singularity is just a log terminal singularity [Ka2, Cor 1.91. For (I) and (3), see [Br, Satz 2.111and [Arl, Theorem 2.71. (1.4). Y is a log del Pezzo surface if the anti-canonical divisor -Ky is Q-ample. Y is a Gorenstein (log) del Pezzo surface if further Y has at worst Du Val singularities (see [Mi, Ch 11, 5.11, [MZl]). A normal variety V is Fano if -Kv is Q-ample. A log del Pezzo surface is nothing but a log terminal Fano surface, and a Gorenstein del Pezzo surface is nothing but a canonical Fano surface. (1.5). Y is a log Enriques surface if the irregularity q ( Y ) = h l ( Y , O y ) = 0 and if m K y N 0 (linear equivalence) for some positive integer m. The smallest m is called the index of Y and denoted by I ( Y ) [Z4, Part I, Definition 1.11. A log Enriques surface of index 1 is nothing but a K3 surface possibly with Du Val singularities. A non-rational log Enriques surface is of index 2 if and only if it is an Enriques surface possibly with Du Val singularities. The case of Y with a unique singularity is classified by Tsunoda [Ts, Proposition 2.21 (see also [Z4, Part I, Proposition 1.61). Proposition. Let Y be a rational log Enriques surface with #(SingY) = 1. Then I ( Y ) = 2 and the unique singularity is of type (1/4n)(1,2n- 1) for some n 2 1. (1.6). The surface Y is relatively minimal if for every curve C , we have either

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2 0 or C2 2 0. Suppose that Y is not relatively minimal. Then there is a curve C such that Ky.C < 0 and C2 < 0. By [MTl, Lemma 1.7 ( 2 ) ] we , see that there is a contraction Y ---f Z of the curve C to a smooth or quotient singularity such that the Picard number p ( 2 ) = p ( Y ) - 1. So every projective surface with at worst quotient singularities has a relatively minimal model. Ky.C

Y is strongly minimal if it is relatively minimal and if there is no curve C with C2 < 0 and C.Ky = 0 [Mi, Ch 11, (4.9)]. (1.7). A smooth projective rational surface X is a Coble surface if 1 - Kxl = 0 while 1 - 2KxI # 0. A Coble surface is terminal if it is not the image of any birational but not biregular morphism of Coble surface. Coble surfaces are classified in [DZ]. Here is an example. Let 2 be a rational elliptic surface with a multiplicity-2 fibre FOand a non-multiple fibre FI of type I , (see [CD] for classification of 2 ) . Let X + 2 be the blow up of all n intersection points in FI. Then X is a terminal Coble surface. Coble surfaces and log Enriques surfaces are closely related. Proposition [DZ, Proposition 6.41. (1) The minimal resolution X of a rational log Enriques surface Y of index 2 is a Coble surface with h o ( X ,-2Kx) = 1 and the only member D in I - 2KxI is reduced and a disjoint union of Di,where Di is either a single (-4)-cumre OT a linear chain with the dual graph below (each Di is contractible to a singularity of type ( 1 / 4 n i ) ( l ,2ni - 1) with ni = #Di):

(-3) - -( -2) - - . . . - -( -2)

-

-( -3).

If we let X t ,

-+ X be the blow up of all intersection points in D , then X t , is a terminal Coble surface with ho(Xt,, -2Kx,,) = 1 and the only member in I -2Kx,, I is a disjoint union of n of (-4)-curues with n = ni. ( 2 ) Conversely, a terminal Coble surface X has a unique member D in I - 2Kx1, and D is reduced and a disjoint union of (-4)-curues.

ci

(1.8). A smooth affine surface S is a Q-homology plane if Hi(S,Q) = 0 for all i > 0. Similarly we can define a Z-homology plane and Q-homology plane with quotient singularities. The following very important theorem is proved by Gurjar, Pradeep and Shastri in their papers [GS], [PSI, [GPS] and GPr].

Theorem 1.9. A Q-homology plane with at worst quotient singularities is a rational surface.


117

Theorem 1.10 [Mi, Theorem 4.101. Let Y be a Q-homology plane. Let u be the number of topologically contractible curves in Y. Then we have: (1) Every topologically contractible curve is isomorphic to the afine line. (2) The Kodaira dimension ,(Yo) = 2 , l or 0,-00 i f and only if I/ = 0 , finite, a,respectively. (Gurjar and Parameswaran [GPa] have determined the number v when &(Yo) = 0). (3) Suppose that Y is a homology plane. Then .(Yo) = 2,1, -00 i f and only i f u = O , l , a , respectively (see (3.17)). SECTION 2. NORMALALGEBRAIC SURFACES Y WITH KODAIRADIMENSION &(Yo)= -aAND FANO VARIETIES

In this section we consider projective varieties with at worst log terminal singularities and Kodaira dimension &(Yo) = -00, where Y o = Y - SingY. The following result is a special case of [MTl, Theorem 2.111. We will sketch a different and direct proof here by making use of [KMM].

Theorem 2.1. Let Y be a relatively minimal surface with at worst quotient singularities. Then one of the following occurs. (1) The Kodaira dimension ,(Yo) 2 0 and K y is numerically effective. (2) .(Yo) = -00 and K y is not numerically effective. To be precise, either (2a) Yo is ruled, i.e. Y o has a Zariski open set of the form P1 x C with a curve C , or (2b) Y is a log del Pezzo surface of Picard number 1. Proof. We may assume that K y is not nef. Then by [KMM, Theorems 4-2-1 and 3-2-11, there is an extremal ray R,o[C] with C a rational curve, and a corresponding morphism @ : Y + Z with connected fibres such that a curve E is mapped to a point by @ if and only if the class of E is in R,o[C].

Case d i m 2 = 2. Then Z has at worst log terminal singularities (= quotient singularities) by [KMM, Proposition 5-1-61, This contradicts the relative minimality of Y . Case d i m 2 = 0. Then PicY is generated over Q by C and hence Picard number p (Y ) = 1 and C is Q-ample. Since K y . C < 0, we have K y = aC (numerically) with a < 0. So the case (2b) occurs. is P1 because F2 = 0 and K y . F < 0 and use genus formula). Clearly the case (2a) occurs. This proves

Case dim Z = 1. Then a general fibre F of (pull back to the theorem.

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Theorem 2.2 ([KMc, Cor. 1.61, [Mi, Ch 11, Theorems 2.1 and 2.171). Let Y be a log del Pezzo surface. Then there is a dominant morphism X o -+ Y o such that X o is an afine-ruled surface (i.e., X o contains a Zariski open set of the form A' x C for some curve C ) . When Y is Gorenstein, Theorem 2.2 was proved in [Z2, Theorem 3.61; the general case of Theorem 2.2 was proved in a lengthy book [KMc, Cor 1.61; Theorems 2.1 and 2.2 together give the proof of the following result, which is the quotient surface case of Miyanishi Conjecture. Theorem 2.3. Let Y be a projective surface with at worst quotient singularities. Suppose that Y is relatively minimal. Then the following are equivalent. (1) K y is not nef. (2) &(YO)= -03. ( 3 ) There is a dominant rnorphism X o -+ Y o such that X o is an afine-ruled surface. Proof. The equivalence of (1) and (2) is proved in [MTl, Theorem 2.111. (2) implies ( 3 ) by Theorems 2.1 and 2.2. Assume (3). Now K ( X ' ) = -03 is clear by considering a ruled surface as a completion of Xo with the boundary equal to the union of a section (or empty set) and a few fibre components. Since K(XO) 2 &(Yo),(2) follows. Now we turn to the topology of smooth locus of a variety. We proposed the following in [Z7]. Conjecture 2.4. Let V be a Fano variety with at worst log terminal singularities. Then the topological fundamental group T I ( V o )is finite.

The affirmative answer to (2.4) would imply the following which was conjectured in [KZ] and is now a theorem of S. Takayama [Ta]. Indeed, (2.4) would imply that nl(V) is finite and we let U + V be the universal cover. Then ~ ( 0 ,=)n X ( O v ) , where n = l r ~ ( V ) l The . Kawamata-Viehweg vanishing implies that x ( 0 x ) = h o ( X ,O X )(= 1) for both X = U and V. Hence n = 1. Theorem 2.5. Let V be a Fano variety with at worst log terminal singularities. Then n l ( V )= (1). The result (2.5) would also follow from the following conjecture which is still open for dimension 4 or higher. It is proved in 3-fold case by [Ca] and [KoMiMo]. Recently, Graber, Harris and Starr [GHS] have proved that any complex algebraic


119

variety having a fibration with rationally connected general fibres and image (or base), is again rationally connected. Conjecture 2.6. Let V be a Fano variety with log terminal singularities. Then V is rationally connected, i.e., any two general points are connected by an irreducible rational curve. Partial answers to (2.4) are given in (2.7)

N

(2.10).

(2.7). When dimension is 2, Conjecture 2.4 was proved in affirmative by [GZl, 21; for a differential geometric proof, see [FKL]. In [KMc, Cor 1.61, it was proved that for a log del Pezzo surface Y,the Y o is rationally connected and hence has finite 7r1(Yo) (see [Ca] and [KoMiMo]). Theorem 2.8 [Z7, Theorem 21. Conjecture 2.4 is true i f one of the following occurs. (1) dimV 5 2. (2) The Fano index r( V ) > dim V - 2 . (3)V has only isolated singularities and r(V) = dim V - 2 = 1. Theorem 2.9 [Z7, Theorem 21. Let V be a Fano variety of Fano index r(V) > dim V- 2 and with at worst canonical singularities. Then nl(Vo) is abelian of order 5 9. Theorem 2.10 [Z7, Theorems 1 and 21. Let V be a Fano variety. Then 7rl(Vo) = (1) if one of the following occurs: (1) The Fano index r ( V ) > dimV - 1. (2) dimV = 3 and V has only Gorenstein isolated singularities.

-

(2.11). The following gives a concrete upper bound for 7rl(Vo) in certain case.

A relation m(Kv + H) 0 in the theorem below occurs when V has Fano index 1 and Cartier index m. It is conjectured that m = 1,2. To prove the theorem

below, we show first that there is a natural surjective map 7rl(Ho) -+ nl(Vo) and also use the fact that H is Du Val K3 or Enriques. Now the theorem follows from the results on H in [KZ, Theorems 1 and 21. For each of the three exceptional cases of ( p , c) below, we note that there is a Du Val K3 or Enriques surface Y with SingY = cAp-1 and 7r1(Yo) infinity [ibid.]. Theorem [KZ, Theorem 31. Let p be a prime number. Let V be a log terminal Fano 3-fold with a Cartier divisor

120


+

H such that m ( K v H ) 0 (linear equivalence) for m = 1 or 2. Suppose that a member H of )HI is irreducible normal and has c singularities of type A,-1 and no other singularities. Then r l ( V o )is soluble; and if ( p , c ) # (2,8), (2, 16), (3,9), then 17rl(Vo)I5 2pk for some k 5 4. N

Remark 2.12. (1) In (2.4) if we replace "log terminal" by "log canonical", then (2.4) has counterexamples; more precisely, if V is a normal Fano surface with at worst rational log canonical singularities, then r l ( V o )contains a finite-index abelian subgroup of rank k ( k = 0,2) [Z8, Theorem 2.31. (2) In (2.9), the upper bound is optimum [MZl, Lemma 61; also "canonical" can not be replaced by "log terminal" [Z3, Appendix]. (2.13). In [Kj2], log del Pezzo surfaces with a unique singularity are classified (including the existence part). The classification of log del Pezzo surface of Cartier index 5 2 were announced in [AN]. In [Kj4], Kojima classified Picard number 1 log del Pezzo surfaces Y of index 2, in a way different from [AN]: there are exactly 18 types of SingY and the r l ( Y o )5 8; the r l ( Y o )= (1) holds if and only if Y contains the f i n e plane as a Zariski open set. (2.14). In [Ni3], the Picard number p ( Y ) of the minimal resolution ? of a log del Pezzo surface Y is bounded from above in terms of the maximum of multiplicities of Y .

SECTION 3. NORMALALGEBRAIC SURFACES Y WITH KODAIRA DIMENSION ,(Yo) = o In this section we consider projective surfaces Y with at worst quotient singularities and Kodaira dimension & ( Y o= ) 0, where Y o = Y - Sing Y. Theorem 3.1 ([Kal, Theorem 2.21, [Mi, Ch 11, 6.1.31). Let Y be a projective surface with at worst quotient singularities. Then the following are equivalent: (1) Y is relatively minimal with Kodairu dimension ,(Yo) = 0. ( 2 ) There is a positive integer m such that m K y 0 (linear equivalence). N

(3.2). The smallest positive integer m with m K y denoted by I = I ( Y ) .

N

0 is called the index of Y and

Proposition 3.3. Let Y be a projective surface with at worst quotient singularities


121

and IKy 0 , where I > 0 is the index of Y. Suppose that Y is irrational. Then one of the following occurs. (1)Y is a (smooth) abelian surface (I = 1) or a hyperelliptic surface (I = 2,3,4,6). ( 2 ) Y has at worst Du Val singularities. The minimal resolution of Y is either a K3 surface (I = 1) or an Enriques surface (I = 2). N

Proof. In notation of (1.3), we have I ( K g + D*) 0. So K(?) 5 0. If K(?) = 0, Then (3.3) follows from the classification of smooth surfaces. If K ( Y )= -03, then ? is an irrational ruled surface over a base curve of genus 2 1. However, K? D*= 0 (numerically) implies that D* contains some horizontal components (this can be seen by going to a relative minimal model of Y) which dominates the base curve and hence is irrational, contradicting the fact that D consists of rational curves only (1.3).So ,(Y) = -03 is impossible. This proves the proposition. N

+

In view of (3.3),to classify those Y with at worst quotient singularities and ,(Yo) = 0, we need only to consider rational surfaces Y with rnKy 0 for some integer rn 2 2 (see (1.5)). These are precisely rational log Enriques surfaces (1.5). N

(3.4). Let Y be a rational log Enriques surface. Then the index I = I(Y) >_ 2. Since IKy 0, there is a canonical Z/(I)-Galois cover 7r : X = Spec@:=: 0(-jKy) + Y which is unramified over (Y \ {non-Du Val singularities}) 2 Yo and satisfies KX 0. Therefore, either X is a (smooth) abelian surface or a K3 surface possibly with some Du Val singularities. Recently, Suzuki [Su] has proved Morrison’s cone conjecture for rational log Enriques surfaces Y : there is a finite rational polyhedral cone which is a fundamental domain for the action of Aut(Y) on the rational convex hull of its ample cone. N

N

Theorem 3.5. Suppose that Y is a rational log Enriques surface of index I and that the canonical Z/(I)-cover X of Y is an abelian surface. Then we have: (1) I = 3 or 5. If I = 3, then Sing Y consists of 9 singularities of type (1/3)(1,1); if I = 5, then Sing Y consists of 5 singularities of type (1/5)(1,2); see [Rel] for notation. (2) For each I = 3,5, there is a unique log Enriques surface YI with X I abelain and I(K) = I . To be precise, YI = X I / ( g I ) , where X3 = Ec3 x Ec3 with Ec3 = C/(Z ZC3) an elliptic curve of period (3 = exp(2~-/3), 93 = diag(C3, &), X5 is the Jacobian surface of the genus-2 curve : y2 = x5 - 1, and g5 in Aut(X5) is induced by the curve automorphism : (2,y ) H (&a,y ) (see [Bl, Example 1.21, [Su, Proposition 1.21 and [Z4, Example 4.21).

+

122


Proof. (1) is proved in [Z4, Theorem 4.11. (2) is proved in [Bl, Su]. Theorem 3.6. Let Y be a log Enriques surface of index I . Then I < 21. Remark 3.7. (1) It is easy to see that the Euler function cp(I) 5 21 and hence I 5 66 [Z4, Part I, Lemma 2.31. In [Bl, Theorem C], it is proved that I 5 21. (2) Examples of Y with prime I ( Y ) are constructed in [Z4, Example 5.3-5.8; Part 11, Example 7.31 and [Bl, Example 4.11.

(3.8). A log Enriques surface Y is maximum if any birational morphism Y --t Z to another log Enriques surface is an isomorphism. By [Z4, Part 11, Theorem 2.11’1, for every log Enriques surface Y of prime index, there is a unique maximum log -+ Y ; Enriques surface Y,,, with I(Ym,,) = I ( Y ) and a birational morphism Y,, each singularity (if exists) of the canonical cover of YmaZis of type A l . The surface Y(A19) in the assertion(3) below is not isomorphic to the unique (modulo projective transformation) quartic K3 surface in P3 with a Dynkin type A19 singularity; neither can Y(D19) be embedded in P3 [KN]. The Y ( D l 9 ) is constructed in two different ways in [Z4, Example 6.11: the and [OZl, Example 11, and Y(A1g)in [Z4, Example 3.2 : the and [OZl, Example 21. The uniqueness problem of Y(D1g) was initiated by [Re2, Round 3, Example 61.

v’]

v]

Theorem [OZ4, Corollary 4; 023, Corollary in $1;OZ1, Theorems 1 and 21. (I) For each I = 13,17,19, there is a unique maximum log Enriques surface Y with I ( Y )= I . (2) All maximum log Enriques surfaces of index 11 form a family of dimension 1 and are all given in [OZ3, $1, Corollary]. (3) For each D in {Dlg,Alg}, there is a unique rational log Enriques surface Y ( D ) whose canonical cover has a singularity of Dynkin type D . The index I ( Y ) equals 3 (resp. 2) when Y equals Y(D1g) (resp. Y(A19)). Next we will investigate the behaviour of 7r1(Yo) and propose a conjecture (3.12) generalizing the one in [CKO]. Note that Z / ( I ) is the image of 7r1(Yo)by a homomorphism.

Theorem 3.9 [Z4, Part 11, Theorem 2.11’, Cor 11. Let Y be a m m i m u m log Enriques surface of odd prime index I . Let X Y be the canonical Z/(I)-cover. Then we have: (1) X has at worst type A1 singularities and #(SingX) 5 6. (2) m ( Y O= ) Z/(O -+


123

Proof. (1) is proved in [Z4, Part 11, Cor. 11. For (2), we have only to show that .rrl(Xo)= (1) because the inverse of Y o via the canonical map X + Y (unrarnified over Y o )is X o with a few smooth points removed (the removal of smooth points in a complex surface does not change T I ) . Since #(SingX) 5 6 by (l),we have .rrl(Xo)= (1) [KZ, Theorem 11. This proves the theorem.

Theorem 3.10 [SZ, Proposition 4.1, Theorem 4.3, Cor 4.41 (1) Suppose that ".rrfg(X0)= (1) + .rrl(Xo)= (1) for all Du Val K 3 surfaces X " holds (X satisfies this condition if (*) in (2) holds for X ) . Let Y be a log Enriques surface of index I . Then either 7rl(Yo)is finite or there is afinite morphism Z + Y from an abelian surface which is unramified over Y o . In particular, .rrl(Yo)contains a finite-index abelian subgroup of rank k (k = 0,4). (2) - Let X + Y be the canonical cover of a log Enriques surface of index I, let X -+ X be the minimal resolution and D = C D i the exceptional divisor. Then .rrl(Yo)= Z / ( I ) if the lattice r = Z[UDi] is primitive in H 2 ( x ,Z) and satisfies: (*) r = r a n k ( r ) 5 18 and the discriminant group r"/r is generated b y k elements with k 5 min{r,20 - r } . Proof. Let X + Y be the canonical Z/(I)-cover. Now the conclusion in (3.10)(1) with Y replaced by X holds by [SZ, Proposition 4.11. So (1) is true for X + Y is unramified over Y o . For (a), [SZ, Theorem 4.31 implies 7r1(Xo)= (1). So (2) is true. This proves the theorem. There is a concrete upper bound of .rrl(Yo)for certain Y .

Theorem 3.11 [GZ3, Theorem 11. Let Y be a rational log Enriques surface of index 2. Assume that Y has no Du Val singularities. Then .rrl(Yo)is a soluble group of order 721722 with ni 5 16. The results (3.9) (3.11) support the following which is just the conjecture in [CKO] when Y is a Du Val K3 surface, i.e., when I ( Y )= 1 and q(Y)= 0. N

Conjecture 3.12. Let Y be a log Enriques surface. Then the universal cover U of Y o is a big open set (= the complement of a discrete subset) of either a Du Val K 3 surface or of C 2 ;in the latter case, U + Y ofactors through a finite etale cover Zo -+ Y o ,where Zo is a big open set of an abelian surface Z . (3.13). Since the canonical cover X + Y of a log Enriques surface is unramified over Y o ,we have .rr1(Yo)/7r1(Xo)= Z / ( I ) . So (3.12) is, in most cases, redcced to the problem of .rrl(Xo) for a Du Val K3 surface X (see (3.5)). We have the

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following:

Theorem [KZ, Theorems 1 and 21. Conjecture 3.12 is true i f Y is a Du Val K 3 or Enriques surface and has several singularities of type Ap-l and no other singularities; here p is a prime number. (3.14). The following results contributes towards an answer in affirmative to (3.12). These are just applications of [CKO, Theorems A and B] to the canonical cover X of Y . Also the upper bound # D 5 15 is an optimum condition for r l ( X o ) to be finite by considering Kummer surfaces. Theorem. (1) Let Y be a log Enriques surface with an elliptic fibration. Then either r l ( Y o )is finite or there is a finite cover of Z -+ Y from an abelian surface which is unramified over Y o . (2) Let Y be a log Enriques surface, X + Y the canonical cover and 2 + X the minimal resolution with D the exceptional divisor. Suppose that # D 5 15. Then r l ( Y o )is finite. (3.15). In [Oh], pairs ( S ,A) of normal surface S and a Q-divisor A satisfying K s + A _= 0 (numerically) are considered. These pairs appear naturally as degenerate fibres in log degeneration; for many interesting cases, he completed the classification of these pairs.

(3.16). In [Kjl, Theorem 0.11, strongly minimal smooth &ne surface S with K(S) = 0 is classified and its invariants are classified (strongly minimal means almost minimal and having no exceptional curve of the second kind [Mi, Ch 11, (4.9)]). In particular, the minimal m > 0 with log pluri-genus P,(S) > 0, the log irregularity q(S) and the Euler number e ( S ) satisfy the following ( r l ( S )is also calculated there, which is generated by at most two elements):

(3.17). In [Fu,$81, all Q-homology planes of Kodaira dimension 0 are classified. It was also proved there that there is no Z-homology plane S of Kodaira dimension K ( S )= 0. The paper [Fu] is very important and also essentially used in [Kjl]. (3.18). Iitaka [I21 conjectured that an f i n e normal variety S is isomorphic to (C*)n if and only if K ( S ) = 0 and q(S) = d i m s . In the same paper, he himself proved it when dim S = 2.


125

According to [Ill, a (possibly open) surface S is logarithmic K 3 if the logarithmic invariants satisfy : q(S) = 0 , p,(S) = 1, K(S) = 0. In [Ill log K3 surfaces were classified. In [Zl], one defines the Iitaka surface as a pair (V,A N ) of smooth projective rational surface V and reduced divisor A N with A K v N 0 and N contractible to Du Val singularities, and the classification of such pairs were done there.

+

+

+

(3.19). In [Kj3], Kojima studies complements S of reduced plane curves with ,(S) = 0; in particular he proves that the logarithmic geometric genus p,(S) = 1. SECTION 4. NORMALALGEBRAIC SURFACES Y WITH KODAIRADIMENSION ,(Yo) = 1 , 2

In this section we consider projective surface Y with at worst quotient singularities and ,(Yo) = 1,2.

(4.1). We first consider the case ,(Yo) = 1. The following is a consequence of [Kal, Theorem 2.31 or [Mi, Ch 11, Theorem 6.1.41. Indeed, in our case, the boundary divisor D* is fractional and contains no effective integral divisor (1.3). Theorem. Let Y be a projective surface with at worst quotient singularities. Suppose that Y is relatively minimal and ,(Yo) = 1. Then there is a positive integer m such that mKy is Cartier and the linear system JmKyI is composed with an irreducible pencil A without base points. Each general member of A is a smooth elliptic curve. So there is an elliptic jibration Y + B .

(4.2). Let Y be a projective surface with at worst quotient singularities. Suppose that Y is relatively minimal and ,(Yo) = 2. Then Ky is nef and big. By [KMM, Theorem 3-1-11] IrnKy) is base point free for m sufficiently divisible, and hence defines a birational morphism 'p : Y -+ 2.This 'p is nothing but the contraction of all curves on Y having zero intersection with Ky. Then 2 has at worst log canonical singularities ([Kal, Theorem 2.91, [Mi, Ch 11, Theorem 4.121). Denote by L C the set of points on Z which is log canonical but not log terminal (i.e., not of quotient singularity). Then we have the following Miyaoka-Yau type inequality proved by [Kbl, 21 and [KNS]. Theorem 4.3. Let Y be a projective surface with at worst quotient singularities. Suppose that Y is relatively minimal and ,(Yo) = 2. Then we have the following, where P runs over all quotient singularities of Z and G p is the local fundamental

126


group at P

(4.4). For smooth and minimal surfaces X of general type, we have Noether inequalities

For singular surfaces, we have:

Theorem [TZ, Theorems 1.3 and 2.101. Let Y be a projective surface with at worst quotient singularities and ,(Yo) = 2. Then the logarithmic geometric genus p g ( Y o ) satisfies the optimum upper bound:

p g ( Y o )< KC

+ 3.

(4.5). For smooth projective surface X of general type, the famous Miyaoka-Yau inequality asserts that cl(X)' 5 3C2(x). Consider log surface (V,D ) with V a smooth projective surface and D a reduced divisor with simply normal crossings. Set Ef = ( K v 0)' and Z2 = c z ( V ) - e ( D ) . Sakai [Sa] proved that ?: 5 3?2

+

provided that D is semi-stable and K(V\ D ) = 2. The following is a lower bound of ?f in terms of '&. These two inequalities together give effective restrictions on the region for non-complete algebraic surfaces V \ D of general type to exist. In the following, (V,D ) is minimal if KV D is nef and there is no (-1)-curve E with E.(Kv D ) = 0.

+

+

Theorem [Z5, Cor. to Theorem C, Theorem D]. Let ( V , D ) be a log surface with D # 0. Assume that (V,D ) is minimal and K(V\ D ) = 2. Assume further that K ( V )2 0 . Then we have

(1) 1 8 c > -& - -. '-15 5 (2) Suppose that p,(V \ D ) 2 3 and IKv DI is not composed with a pencil. Then -2

+


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SECTION5. AUTOMORPHISMS OF ALGEBRAIC SURFACES - SMOOTH SURFACE CASE

(5.1). We mention some background of Aut(X) where X is a smooth projective rational surface. Aut(X) had been studied by S. Kantor more than one hundred years ago [Kt]. It was continued by Segre, Manin, Iskovskih, Gizatullin and many others [Se], [Mal, 21, [Is], [Gi]. See also [Hol], [ H o ~ ]In . [DO], the group of automorphisms of any general del Pezzo surface is described and it turns out that its discrete part is equal to the kernel of the Cremona representation on the moduli space of n points in P2. Very recently, de Fernex [dF] constructed all the Cremona transformations of P2 of prime order, where he employed the methods different from those used by Dolgachev and Zhang in [ZD]. In [ZD], minimal pairs ( X , G ) with prime order p = IGI was considered. In particular, using the recent Mori theory, it was shown there that if the G-invariant sublattice of Pic X has rank 1 then p 5 5 unless X = P2;the short and precise classification of these pairs, modulo equivariant isomorphism, was also given there. Generic Enriques surfaces have infinitely many automorphisms. Those Enriques surfaces with finite automorphisms have been classified by S. Kond6 ([Kon4], see also [Ni2]); there are seven families of such Enriques surfaces. It follows that K3 covers of Enriques surfaces all have infinite automorphism groups.

For K3 surfaces, much progress on their automorphism groups has been done by Kond6, Keum, Dolgachev, Oguiso, and Zhang ([Konl, 2, 31, [Kel, 31, [KK], [DK], [OZl, 3, 4, 51). It is known that minimal surfaces of general type has only finite automorphism groups. It was Xiao [Xil] who gave a proof of the existence of a bound for the order of the automophism group, which is linear in the Euler number of the surface. For curves C of genus 2 2, a classical theorem of Hurwitz gave a sharp bound JAut(C)J5 84(g(c) - 1) = -42e(C).

Theorem [Xi2, Theorem 21. Let X be a minimal surface of general type. Then IAut(X)I 5 ( 4 2 K ~ ) with ~ , equality if and only if X Z (C x C ) / N , where C is a curve with (Aut(C)(= 84(g(C) - l), N a normal subgroup of Aut(C x C) acting freely on C x C and preserving the two projections of C x C. ( 5 . 2 ) . In [ZD], pairs (X,G) of a smooth rational projective surface X and a finite group of automorphisms are considered. A pair is minimal if every G-equivariant birational morphism to another pair (2,G) is an isomorphism.

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Theorem [ZD, Therorem 11. Let (X,p p ) be a minimal pair with p a prime number. ( 1 ) If the invariant sub-lattice (PicX)f’p has rank at least 2, then X is a Hirzebmch surface and the pair (X,p p ) is birationally equivalent to a pair (P2,p p ) . (2) If (PicX)f’p has rank 1, then the pairs are classified in [ Z D , T h e o r e m l ] ;in particular, we have p 5 5 unless X = P2.

(5.3). Let X be a K3 surface. The following are well known (cf. [BPV]). (1) H y X ) = c w x , where wx is a nowhere vanishing global holomorphic 2-form on X. (2) H2(X,Z) is an even unimodular lattice of signature (3,19) with the cup product, so we have an isomorphism

where U (resp. &) is the even unimodular lattice of signature ( 1 , l ) (resp. (8,O)). (3) Pic(X) is isomorphic to the NQron-Severi group NS(X), and hence can be viewed as a sublattice of H2(X,Z). The rank of Pic (X), called the Picard number of X, is denoted by p(X). This number can take the value 0, 1, ..., 20. The lattice Pic (X) is hyperbolic(=Lorentzian) if X is projective, and is semi-negative definite or negative definite if X is not projective. (4) Recall that all K3 surfaces are Kahler [Siu], so Hodge decomposition holds for them. ( 5 ) Let C(X) c H’”(X,R) := H’>’(X)nH2(X,R) denote the Kahler cone of X, the set of all classes of symplectic forms of KahlerEinstein metrics on X. In K3 surface case, C ( X ) can be numerically characterized as follows:

C(X) = {w E H1il(X,R): (w,w) > 0, (w, R) > 0 for all smooth rational curves R) For a compact Kahler manifold, Nakai-Moishezon type criterion, i.e. the characterization of the Kahler cone, is highly non-trivial (see [DP]).

( 6 ) Let Then

T

be an element of Pic ( X ) with (r,r ) = -2d (d > 0), and z -+ z

+ (z,r ) r / d

defines an isometry of Pic (X), called a (-2d)-reflection.

( T , Pic

(X)) c dZ.


129

Let W(Pic(X)) (resp.W(Pic(X))(2))be the subgroup of the orthogonal group O(Pic(X)) generated by all reflections (resp. all (-2)-reflections). These are normal subgroups of O(Pic (X)) and, by linearity, acts naturally on H1?l(X,R). The set { w E H1ll(X,R): ( w , w ) > 0}

has two components, each a cone over a 19-dimensional hyperbolic manifold with constant curvature. C(X) is contained in one of the two components, and the action of W(Pic (X))(2)on this component has C(X) as its fundamental domain.

(7) If X is projective, the ample cone D ( X ) := C(X) fl Pic (X) 8 R is non-empty and can be numerically characterized as D ( X ) = { w E Pic (X)8 R : (w, w ) > 0, ( w , R) > 0 for all smooth rational curves

R). The group W(Pic (X))(2)acts on the component P+(X) of {w E Pic (X) 8 R : ( w , w ) > 0)

containing D(X), and has D ( X ) as its fundamental domain. Note that O(Pic (X))acts on P+(X) and is a semi-direct product of the normal subgroup W(Pic (X))(2)and the symmetry group SymD(X) of the cone D ( X ) , i.e. O(Pic (X))/W(Pic ( x ) ) ( ~s )~ ~ D ( x ) . (5.4). The Torelli theorem asserts that a K3 surface is determined up to isomorphism by its Hodge structure. More precisely we have:

Theorem ([PSS],[BR]).Let X and Y be K3 surfaces, and let

4 : P ( X ,Z)

+ H 2 ( Y , Z)

be an isometry. Extend 4 to H 2 ( X ,C ) or to H2(X,R) b y tensoring with C or R. Then : (1) I f 4 sends H2>'(X)to H2>'(Y),then X and Y are isomorphic. (2) If 4 also sends C(X) to C ( Y ) ,then 4 = f * for a unique isomorphism f : Y +

X.

130


(5.5). Let X be a projective K3 surface. Torelli theorem shows that there is a map Aut(X) -+ O(Pic (X))/W(Pic (X))(2)2 S y m D ( X ) which has finite kernel and cofinite image. So in practice, if we want t o describe Aut(X), the main step is to calculate O(Pic (X))/W(Pic (X))(2).This is in general a highly nontrivial arithmetic problem, if the group is infinite. There are 3 cases: (1) W(Pic (X))(2)is of finite index in O(Pic ( X ) )

(2) W(Pic (X))(2) is of infinite index in O(Pic (X)), but W(Pic ( X ) ) is of finite index in O(Pic ( X ) ) . (In this case, we call Pic ( X ) reflective.) (3) W(Pic (X)) is of infinite index in O(Pic (X)),i.e. Pic (X) is not reflective.

Remark. The case (1) occurs if and only if Aut(X) is finite. If p(X) 2 3, this occurs if and only if X contains at least one but finitely many smooth rational curves. If p(X) = 2 , this occurs if and only if X contains a smooth rational curve or an irreducible curve of arithmetic genus 1, if and only if Pic (X) represents -2 or 0 [PSS]. Nikulin [Nil, Ni4] and Vinberg classified all such lattices of rank 2 3 belonging to the case (1). It follows from the classification that every algebraic Kummer surface has an infinite automorphism group (cf. [Ke2]). The classification of the Nhron-Severi lattice is also utilized in [Ogl - Og3], where he has proved the density of the jumping loci of the Picard number of a hyperkahler manifold under small 1-dimensional deformation, where he reveals the structure of hierarchy among all the narrow Mordell-Weil lattices of Jacobian K3 surfaces. ( 5 . 6 ) . For finite groups which can act on a K3 surface, the following results are given by S. Mukai and S. Kondo.

Theorem [Mu],[Kon2]. Let X be a K 3 surface and let G be a finite symplectic subgroup of Aut(X), i.e. G acts trivially on H2io(X). Then G is isomorphic to a subgroup of the Mathieu group M23, which has at least five orbits on a set 0 of 24 elements. In particular, IGI 5 960. Theorem [KonS]. The maximum order among all finite groups which can act on a K3 surface is 3840, and is uniquely realized by the group (Z/2Z)4. A 5 .2 / 4 2 acting on the Kummer surface Km(EJ=T x E-), where €3is the elliptic curve with

fl

as its fundamental period.

Some projective K3 surfaces, including all algebraic Kummer surfaces and K3 covers


131

of Enriques surfaces, have infinite automorphism groups. Given a projective K3 surface X with Aut(X) infinite, it is an interesting problem to determine a set of geometric generators of Aut(X). This problem has been settled for certain classes of K3 surfaces. These results are given in ( 5 . 7 ) - (5.10) below. (5.7). Two most algebraic K 3 surfaces Vinberg[Vin] calculated Aut(X) for two K3 surface with transcendental lattice

respectively. In both cases, the full reflection group W(Pic (X)) is of finite index in O(Pic (X)). (5.8). generic Jacobian Kummer surfaces

Let C be a smooth curve of genus 2. The Jacobian variety J ( C ) of C is an abelian surface with a natural involution r and the quotient variety J ( C ) / r has 16 singularities of type A l . This surface can be embedded as a quartic surface F in P3 with 16 nodes. The minimal resolution X of J ( C ) / r is called the Jacobian Kummer surface associated with C. We call X generic if the NBron-Severi group of J ( C ) is generated by the class of C. For X generic, the transcendental lattice T(X) can be computed as follows:

T(X) = U ( 2 )@ U ( 2 ) @< -4 > . Note that Aut(X) is isomorphic to the birational automorphism group B i r ( F ) of the singular quartic surface F . At the last century it was known that X has many involutions, that is, sixteen translations induced by those of J ( C ) by a 2-torsion point, sixteen projections of F from a node, sixteen correlations by means of the tangent plane collinear to a trope, and a switch defined by the dual map of F . In 1900, Hutchinson found another 60 involutions associated with Gopel tetrads. Since Hutchinson, for generic X no other automorphism had been provided until new 192 automorphisms were given in [Kel].

Theorem [Kel]. For a generic Jacobian Kummer surface, there are 192 new automorphisms of infinite order which are not generated by classical involutions. Theorem [Konl]. The automorphism group of a generic Jacobian Kummer surface is generated by the classical involutions and the 192 new automorphisms. Theorem [Ke3]. For F generic, all birational automorphisms of F are induced by Cremona transformations of P3.

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(5.9). Kummer surfaces associated with the product of two elliptic curves The following four cases were considered. In each case, a set of generators of Aut(X) is given in [KK].

Case I. X = K m ( E x F ) where E and F are non-isogenus generic elliptic curves. Case 11. X = K m ( E x E ) where E is an elliptic curve without complex multiplications. Case 111. X = Km(E, x E,) where w is a 3rd root of unity and E, is the elliptic curve with T as its fundamental period. Case IV. X = K m ( E a x E-). The transcendental lattice T ( X ) can be computed as follows: U(2) @ U(2), (Case I); U(2)B < 4 >, (Case 11);

Remark. In Case I the group W(Pic (X)) is of finite index in O(Pic (X)) and in other cases not. (5.10). Quartic Hessian surfaces Let S : F(zo, ~ 1 ~ x 2=, 0~be) a nonsingular cubic surface in P3. Its Hessian surface is a quartic surface defined by the determinant of the matrix of second order partial derivatives of the polynomial F . When F is general enough, the quartic H is irreducible and has 10 nodes. It contains also 10 lines which are the intersection lines of five planes in general linear position. The union of these five planes is classically known as the Sylvester pentahedron of S . The equation of S can be written as the sum of cubes of some linear forms defining the five planes. A nonsingular model of H is a K3 surface I?. Its Picard number p satisfies the inequality p 2 16. Note that A u t ( k ) 2 B i r ( H ) . In [DK] an explicit description of the group B i r ( H ) is given when S is general enough so that p = 16. In this general case, the transcendental lattice T ( H )can be computed as follows:

T ( H ) U @ U(2) @ A2(-2). 1


133

Although H , in general, does not have any non-trivial automorphisms (because S does not), the group B i r ( H ) F Aut(fi) is infinite. It is generated by the automorphisms defined by projections from the nodes of H , a birational involution which interchanges the nodes and the lines, and the inversion automorphisms of some elliptic pencils on 13. This can be compared with the known structure of the group of automorphisms of the Jacobian Kummer surface (5.8). Indeed, the latter surface is birationally isomorphic to the Hessian H of a cubic surface [Hu] but the Picard number of fi is equal to 17 instead of 16. (5.11). Let us explain the method for computing the automorphism group of an algebraic K3 surface, which was first employed by S. Kond6 for generic Jacobian Kummer surface case (5.8)[Konl]. The two cases (5.9) and (5.10) use this method, and even the first case (5.7) can also be calculated by the same method (See [Bor2]). Let X be an algebraic K3 surface with large Picard number, say p ( X ) 2 3. Suppose that Aut(X) is infinite. Then the ample cone D ( X ) is not a (finite) polyhedral cone, i.e. has infinitely many faces. Hence, it is difficult to describe D ( X ) explicitly. Assume that one can find a polyhedral cone D' in D ( X ) ,

a set of automorphisms {ga} of X whose action on D ( X ) has D' as a fundamental domain. Then, by (5.5), one can conclude that the automorphisms {ga} generate the whole group Aut(X), up to finite groups. In addition to {ga}, some symmetries of D' (not all elements of SymD' in general) may realize as automorphisms of X and some projectively linear automorphisms, if any, generate the kernel of the map in (5.5).

Remark. In the above, ga corresponds to a face of D' orthogonal to a vector a , and acts on D ( X ) like a reflection, i.e. sends one of the half-spaces defined by a to the other half-space defined by a or to one of the two half-spaces corresponding to gL1. The second case actually occurs in generic Jacobian Kummer surface case (5.8). (5.12). To find such a polyhedral cone D' c D ( X ) ,Kond6 used the known structure of the orthogonal group of the even unimodular lattice 111,25 of signature (1,25). (Such a lattice is unique up to isomorphism and is isomorphic to A @ U ,

134

JONGHAE KEUM AND DE-$1 ZHANG

where A is the Leech lattice, i.e. the even unimodular negative definite lattice of rank 24 which contains no vectors of norm -2.) To be more precise, the following steps lead to the calculation of Aut(X).

Step 1. Compute Pic ( X ) .

Step 2. Embed Pic ( X ) primitively into I I 1 , 2 5 = A @ U such that the projection of the Weyl vector w = (0, (0,l)) E A @ U onto Pic ( X )8 R must be an ample class. Conway [Co] described a fundamental domain D of the reflection group W ( I 1 1 , 2 5 ) (in~ )terms of the Leech roots (=roots with intersection number 1 with is generated by the Weyl vector w ) . More precisely, he showed that w(111,25)(2) (-2)-reflections corresponding to Leech roots.

Step 3. The fundamental domain D of the reflection group W ( 1 1 1 ~ 5 )cuts ' ~ ) out a finite polyhedral cone D' inside the ample cone D ( X ) . In other words,

D' = D n P + ( X ) , where P + ( X ) is the positive component of {w E Pic ( X )@ R: (w,w) > 0). Indeed, D contains the Weyl vector w and, by Step 2, the projection of w is contained in D'. Determine the hyperplanes a which bound D'. The reason why D' is polyhedral comes from Borcherds [Borl];among infinitely many faces of D , those intersecting P + ( X ) bound D', and these faces correspond to Leech roots having a non-zero projection onto Pic ( X )8 R.

Step 4. Match the faces a of D' with automorphisms ga such that ga sends one of the half-spaces defined by a to the other half-space defined by a or to one of the two half-spaces corresponding to g G 1 . This allows one to prove that the automorphisms ga generate a group of symmetries of D ( X ) ,having D' as its fundamental domain. Step 5. Take care of SymD'. See if which symmetries of D' realize as automorphisms of X . Finally see if there are any projectively linear automorphisms, (which generate the kernel of the map in ( 5 . 5 ) ) .

Remark. In the known cases (5.7) - (5.10), the embedding Pic ( X ) c A @ U is given in such a way that Pic ( X ) is the orthogonal complement of a root sublattice of A @ U . For example, in case (5.7) the orthogonal complement of Pic ( X ) in


135

A @ U is a primitive sublattice of rank 10 which contains a negative definite root lattice of type As A:, which is of index 2. In practice, Step 4 seems most complicated. The automorphism ga works like a reflection, but is not necessarily an involution. It may be of infinite order. At any rate, ga may be geometrically evident, or can be picked up from a list of already known automorphisms, or may be found by looking at extra structures of X , e.g. elliptic fibrations, double plane structures, ..., etc. In worst cases, one has to find an (abstract) effective Hodge isometry of H 2 ( X ,Z) and then realize it geometrically. The reason why the beautiful combinatorics of the Leech lattice plays a role in the description of the automorphism groups of K3 surfaces is still unclear to us. We hope that the classification of all K3 surfaces whose Picard lattice is isomorphic to the orthogonal complement of a root sublattice of I J 1 , 2 5 will shed more light to this question.

+

Remark (5.13). The method (5.12) also works for some supersingular K3 surfaces. I. Dolgachev and S. Kondo [DoKo] have computed the automorphism group of a supersingular K 3 surface in characteristic 2 whose Picard lattice is U @ Dz0. (For supersingular K3 surfaces Torelli type theorem holds, i.e. an automorphism of a supersingular K3 surface is determined by its action on the Picard lattice.) An even lattice L is reflective if its reflection group W ( L )is of finite index in O ( L ) (see (5.5)). The lattice U@D20 is reflective as it is pointed out by Borcherds [Borl] and it is the only known example, up to scaling, of an even reflective hyperbolic lattice of rank 22. The range of possible rank of an even reflective lattice of signature (1,. - l ) , r 2 1, is given by Esselmann [Es]: it takes the same range 1, 2, ..., 20, 22 as the Picard number of a K3 surface in positive characteristic. SECTION 6. AUTOMORPHISMS OF ALGEBRAIC SURFACES - SINGULAR SURFACE CASE

(6.1). In [MM] and [MZ3], one considers pairs (V,G) of surface V and group G of automorphisms, where V may be singular and even non-complete. (6.2). Let Y be a Gorenstein del Pezzo singular surface of Picard number 1. In [Z9], one classifies all actions on Y by cyclic groups Z / ( p ) of prime order p 2 5. Theorem [Z9, Theorems A and C]. Let Y be a Gorenstein del Pezzo singular surface of Picard number 1. Then we have: (1) Either JAut(Y)I= 2"3' for some 1 5 a + b 5 7, or Aut(Y) 2 Z / ( p ) for every prime p 2 5 and hence IAut(Y)I = 00.

136


(2) (Aut(Y)I is finite i f and only i f either Sing Y = A7 or K$ = 1 and I - Kyl has at least three singular members. (3) Let p 2 5 be a prime. Suppose that Y is not isomorphic to the quadric cone in P3. Then modulo equivariant isomorphism, there is either none, or only one, or exactly p 1 action(s) of Z/(p) on Y. All actions are given in [Z9].

+

Remark 6.3. We like to compare (6.2) with known results for smooth del Pezzo surfaces. (1) If X i s a generic rational surface with K i 5 5, then IAut(X)I divides 5! (see [DO], [Kill . (2) If X is a del Pezzo surface of degree 3, 4, then Aut(X) does not contains Z/(p) for any prime p 2 7, and modulo equivariant isomorphism there is at most one non-trivial Z/(5) action on X [Hol, 21. (3) Let X be a rational projective surface with a non-trivial Z/(p)-action for some prime p such that the Z/(p)-invariant sublattice of Pic X is of rank 1 (this condition is automatic if the Picard number p ( X ) = 1). If X is smooth, then p 5 5 unless X = P2[ZD, Theorem 11. ACKNOWLEDGEMENT We would like to thank the referee for careful reading and suggestions which improve the paper. This article was initiated when both of us were attending the conference - Algebraic Geometry in East Asia - in August 2001, and we would like to thank the organizers for providing the wonderful environment and the encouragement for us to write a survey article. This work was finalized when the secondnamed author visited Korea Institute for Advanced Study in December 2001, and he likes to thank the institute for the hospitality. This work was partially supported by Korea Science and Engineering Foundation (R01-1999-00004) and an Academic Research Fund of National University of Singapore. REFERENCES

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[Se] B. Segre, The Non-singular cubic surfaces, Oxford University Press, Oxford, 1942. [SZ] I. Shimada and D. -9. Zhang, Classification of extremal elliptic K 3 surfaces and fundamental groups of open K3 surfaces, Nagoya Math. J. 161 (2001), 23-54. [Su]K. Suzuki, On Morrison’s cone conjecture for Klt surfaces with Kx 0, Comment. Math. Univ. St. Paul 50 (2001), 173-180. [Siu] Y . T . Siu, Every K 3 surface is Kiihler, Invent. Math. 73 (1983), 139-150. [Ta] S. Takayama, Simple connectedness of weak Fano varieties, J. Algebraic Geom. 9 (2000), 403-407. [Ts] S. Tsunoda, Structure of open algebraic surfaces, I, J. Math. Kyoto Univ. 23 (1983), 95-125. [TZ] S. Tsunoda and D. -&. Zhang, Noether’s inequality for noncomplete algebraic surfaces of general type, Publ. Res. Inst. Math. Sci. 28 (1992), 21-38. [Vin] E.B. Vinberg, The two most algebraic K3 surfaces, Math. Ann. 265 (1985), 1-21. [Xil] G. Xiao, Bound of automorphisms of surfaces of general type. I, Ann. of Math. (2) 139 (1994), no. 1, 51-77. [Xi21 G. Xiao, Bound of automorphisms of surfaces of general type. 11, J. Algebraic Geom. 4 (1995), no. 4, 701-793. (Zl] D. -&. Zhang, On Iitaka surfaces, Osaka J. Math. 24 (1987), 417-460. [ZZ] D. -Q. Zhang, Logarithmic del Pezzo surfaces of rank one with contractible boundaries, Osaka J. Math. 25 (1988), 461-497. [Z3] D. -&. Zhang, Logarithmic del Pezzo surfaces with rational double and triple singular points, Tohoku Math. J. 41 (1989), 399-452. 1241 D. -&. Zhang, Logarithmic Enriques surfaces, I; 11, J. Math. Kyoto Univ. 31 (1991), 419-466; 33 (1993), 357-397. [Z5] D. -&. Zhang, Noether’s inequality for noncomplete algebraic surfaces of general type, 11, Publ. Res. Inst. Math. Sci. 28 (1992), 679-707. [ZS] D. -&. Zhang, Algebraic surfaces with nef and big anti-canonical divisor, Math. Proc. Cambridge Philos. SOC. 117 (1995), 161-163. (271 D. -&. Zhang, The fundamental group of the smooth part of a log Fano variety, Osaka J. Math. 32 (1995), 637444. [Z8] D. -&. Zhang, Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts, Trans. Amer. Math. SOC.348 (1996), 4175-4184. [Z9] D. -&. Zhang, Automorphisms of finite order on Gorenstein del Pezzo surfaces, Trans. Amer. Math. SOC.354 (2002), 4831-4845. [ZD] D. -&. Zhang, Automorphisms of finite order on rational surfaces, with a n appendix by I. Dolgachev, J. Algebra 238 (2001),560-589.

=

J. Keum Korea Institute for Advanced Study 207-43 Cheongryangri-dong, Dongdaemun-gu

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Seoul 130-012, Korea Email : [email protected] D. -&. Zhang Department of Mathematics National University of Singapore 2 Science Drive 2, Singapore 117543 Republic of Singapore E-mail: [email protected]

LINEAR SERIES OF IRREGULAR VARIETIES JUNGKAI A. CHEN (JOINT WITH CHRISTOPHER D. HACON)

1. INTRODUCTION Let X be a nonsingular projective variety over complex number C of dimension n and D be a divisor on X . The linear series ID1 is defined to be the set of effective divisors which are linear equivalent t o D. It's natural t o ask under what circumstance the linear series ID1 is non-empty or base point free. Or one might even want to know if (DI defines a barional map or an embedding. Regarding the base point freeness of linear series, F'ujita conjectured the following:

+

Conjecture 1.1. If L is an ample divisor on X , then (Kx m L ( is free f o r m n + 1.

2

This conjecture can be easily proved in dimension 1. For dimX = 2,3,4, the statement is verified by Reider [Re], Ein and Lazarsfeld [ELl], and Kawamata [Ka] respectively. For higher dimensional varieties, a remarkable breakthrough is made by Angehrn and Siu [AS],which assert that IK,y +mL1 is free for m 2 i ( n 2+n) 1. Their proofs use various form of vanishing theorem essentially. The purpose of this note is to introduce some application of Fourier-Mukai transform in the study of linear series. To setup, let A be an abelian variety and 3 be a coherent sheaf on A. 3 is asaid to be I T o if H i ( A ,.F@P) = 0 for all P E Pico(A) and i > 0. By using Fourier-Mukai transform, it's easy to see the following results.

+

Lemma 1.2. If 3 # 0 is ITo, then ho(A,3)# 0 . Lemma 1.3. If F as ITo and there as a surjective map 3 + k ( y ) , then the induced map H o ( A ,F @ P )-+ H o ( A ,k(y)@P) is surjective for general P . Consider now a variety with q ( X ) > 0. Let alb : X -+ Alb(X) be the Albanese map. And let a ( X ) be the dimension of image of alb. We say X is of maximal Albanese dimension if a ( X ) = dimX. In general, our idea here can realized as following: In order to study the linear series ID\, we consider the push-forward 3 := alb,O(D). If 3 satisfied ITo, then one can obtain some information on base points by Lemma 1.3. For example, we are able to show the following result.

Theorem 1.4. IfdimX - a ( X ) 5 2, then )7K,y(defines a birational map. 2000

MSC. 14C20, 14K12

143

JUNGKAI A. CHEN (JOINT WITH CHRISTOPHER D. HACON)

144

In particular, for a %fold with q > 0, the 17Kx( defines a birational map. 2. FOURIER-MUKAI TR.ANSFORMS

In this section, we are going to describe the Fourier-Mukai transform (cf. [Mu]) briefly. And then we are going to demonstrate some easy application of FourierMukai transform. In particular, we prove a generalization of Lefschetz base point freeness theorem to coherent sheaves. This generalization turns out to be useful in the study of irregular varieties. Let A be an abelian variety, and denote the corresponding dual abelian variety by A . Let P be the normalized Poincar6 bundle on A x A. For any point y E A, let P - PJAxfy) denote the associated topological trivial line bundle. Define the -.. functor of OA-modules into the category of Oa-modules by

s

S ( M ) = 7r'a,*(P@npq. The derived functor RS of S then induces an equivalence of categories between the two derived categories D ( A ) and D ( A ) . In fact, by [Mu]: There are isomorphisms of functors: RS 0 d (-1A)*[-g]

and

d Rs 2 (-1a)*[-g], 0

where [-g] denotes "shijt the complex g places to the right". A coherent sheaf 3 on A is said to be I T i if there exists an integer i ( 3 )such that for all j # i ( F ) ,@(A, F @ P )= 0 for all P E Pico(A). It is easily seen that if 3 satisfies the I T i , then RjS(F)= 0 for all j # i ( F ) . We will denote the coherent sheaf Ri(*)S(3)on A by y and call it the Fourier-Mukai transform of 3. We remark that that is a map R j S ( F ) @ k ( y )-+ Hj(A,F@Py).

Lemma 2.1. If F # 0 is I T i , then H i ( A , F )# 0. is a locally free sheaf of rank h i ( A , 3 ) . If h i ( A 1 3 )= 0, then Proof. follows that 3 = (-1)*d = 0. Which is a contradiction.

9 = 0.

It

Definition 2.2. Let 3 be a coherent sheaf on an irregular variety X . F is said to have an essential base point at y i f there is a surjective map 3 k ( y ) such that for all P E Pico(X), the induced map H o ( X ,3 @ P ) H o ( X ,k(y)) is zero. --$

--f

Proposition 2.3. If 3 is a non-zero sheaf on an abelian variety satisfying I T o , then F has no essential base points.

LINEAR SERIES OF IRREGULAR VARIETIES

145

Proof. Suppose that F has an essential base point at y. Let q5 # 0 be a surjective +#O map as above, .Fy := k e r ( F -+ k(y)). Then the exact sequence

0 --+ Fy@P+ F@P + k(y)

+

0

gives

Ho(A,Fy@P) 5 Ho(A,.F@P), Ho(A, k(y)) 5 H1(A, .Fy@P), for all P E Pico(A). Computing the Fourier-Mukai transform, one has

0

-+

R0S(Fy) -+ ROS(F) -+ Py-+ R1S(FY) 0. --f

By Grauert's theorem, V := Ros(.F) and R o s ( F y )are vector bundles of the same rank, and RIS(Fy)is a line bundle. If follows that Py-+ R1s(.Fy)is an isomorphism and hence V + Py is the zero map. Therefore, one has a triangle

R'S(.Fy)

-

R'S(F) -2 R'S(k(y))

--

in D ( A ) . There is a second triangle in D ( A ) given by (cf. [Ha] TR2)

R'S(F) -% R'S(k(y))

R's(Fy)[l]

which is easily seen to be isomorphic to (cf. [Ha] TR3)

v 2 R'S(7) 5Py= R*S(k(y))

Pya3 V[1]. Taking the Fourier-Mukai transform again, one has .Fy .F k(y)[l] in D(A). 0 This is the required contradiction. --f

Corollary 2.4. Let L be an ample line bundle on an abelian variety A. Then 12LI is base point free. Proof. L is ITo and there is a surjective map L k(y) for any given point y E A. Thus for general P E Pico(A), Ho(A,L g P ) + Ho(A, k(y)) is surjective. Which means that y is not a base point of IL PI for general P. One can pick P so that y is not a base point of IL PI and IL - PI. Hence y is not a base point of 12L(. 0 ---f

+

3. ADJOINTLINEAR

+

SERIES OF IRREGULAR VARIETIES

In order to apply the criterion for essential base point freeness to linear series of irregular varieties, we will need the following result:

Theorem 3.1. Let X be a smooth projective varieties with q ( X ) > 0. Let a : X --+ A be a non-constant morphism to an abelian variety A. Let L be a lane bundle on X and fi. a point x E X . Suppose that i) hi(A,a,L@P) = 0 for all i > 0 and P E Pico(A). ii) a,(L@T,) # a,(L).

146


Then x is not an essential base point of JLI. I n particular, x is not a base point of 1 2 4 .

Proof. For a fixed point x in X, we start by considering the sequence 0 -+ L a , and its push-forward via a : X

o

-+

a,(LBZ,)

3

---f

L

---$

k(x) + 0 ,

A

+ a,(L) + a,(k(z)>-+

Rla,(LBZ,)

+.

...

By assumption a,(L@Z,) + a,(L) is not an isomorphism. It is clear that a,(k(x)) = k ( y ) where y = a(.). Let Q be the kernel of k ( y ) R1a,((L)@Z,). It follows that Q # 0. However, k(y) has no non-zero sub-sheaf other than itself. Hence Q = k ( y ) . We therefore have an exact sequence: --f

0

-+

a,(LBZ,)

--+

a,(L)

+ k ( y ) + 0.

a,(L) satisfies the I.T. of index 0. By Proposition 2.3, y is not an essential base point of a,(L). Therefore, x is not a base point of L@a*Q for some Q E Pico(A). By the semi-continuity of h o ( X ,L@Z,@a*Q) as Q varies in Pico(A), it follows that x is not a base point of IL a*Ql for general Q E Pico(X), and hence the theorem. 0

+

As an application, we have the following result.

Corollary 3.2. Let X be a variety of maximal Albanese dimension, and L a nef and big line bundle on X. Then B s l 2 K x 2L( is contained in the exceptional locus of a : X + Alb(X).

+

Proof. Let a : X -+ A be the Albanese map. We first note that a,(wx@L) is a sheaf satisfying I T o (cf. [KO] Corollary 10.15 and [HP] Lemma 2.7). Clearly, it is non-zero since a is generically finite onto its image. Let x E X be a point which is not in the exceptional divisor. There is an open neighborhood U of x such that a : U --+ a ( U ) is a finite morphism. Hence condition ii) in Theorem 3.1 is satisfied and we are done. 0 One can generalize this result to a more general setting.

Theorem 3.3. Let a : X + Z := albx(X) c A be the Albanese morphism of a nonsingular projective variety X to a subvariety Z contained in an abelian variety A. Let L be an ample line bundle on X , e := dim(X) - dim(Z) and m := Let U be an open subset of Z and V := a - l ( U ) such that alV : V + U is a smooth morphism. Then IKx mLI has no essential base points contained in U . I n particular B s l 2 K x 2mLI c X - V .

w.

+

Proof. See [CHI.

+

I7

147


4. NON-VANISHING OF ADJOINTLINEARSERIES For a non-zero sheaf satisfying ITo, we have seen the non-vanishing of its cohomology. This leads to a non-vanishing of linear series in a natural way.

Lemma 4.1. Let X be a variety with q ( X ) > 0 and L be a nef and big divisor, And a : X -+ A is a non-trivial morphism to an abelian variety A with general fiber F . Then h o ( X ,K x L ) # 0 if and only if ho(F,K F L F ) # 0. Proof. Let F := alb,(C?(Kx + L ) . It's easy to check that F is ITo. By Lemma 1.1, one sees that h o ( X ,K x + L ) # 0 if and only if F # 0. We are done since F # 0 if and only ho(F,K F L F )# 0. 0

+

+

+

Theorem 4.2. Let X be a variety with q ( X ) > 0. Suppose that dimX - a 5 2. Let L be a nef and big divisor such that K x + L is nef. Then H o ( X ,K x L ) # 0.

+

Proof. By the previous Lemma, it suffices to show the statement for X,. The assertion is true for 1-dimensional varieties. If X , has dimension 2 with q(X,) > 0, then we proceed inductively. If X , has dimension 2 with q(X,) = 0, then we use Riemann-Roch theorem. 1 0 h ( ~ ~ , ~ F , ~ ~ ~ , ) = X ( ~ ~ , ~ F2 , + ~ X , ) Y =-> ~I . +

0

This completes the proof.

5. PLURICANONICAL MAPS In this section, we are going to show the application t o the study of linear series. Let X be a smooth projective variety and 2 = albx(X) c Alb(X) its Albanese image.

Lemma 5.1. Let g : X -+ S be a suryective morphism of smooth projective varieties, L a nef and big Cartier divisor on S . Let X, be a general fiber of X + S. For any integer m 1 2, assume that Pm(X,)> 0 and write JmKxsI= JHm( + F" where IHml has n o fixed divisors. Assume furthermore that K X is S-big and K ( S ) 2 0 . Then, after replacing X by an appropriate birational model, there exists a n integer b > 0 (any suficiently big and divisible integer b > 0 will do) such that for any suf3ciently big and divisible integers a > 0 , there is a divisor B E lab(m - 1 ) K x - bg*L(, B has normal crossings support and Ix, < Ix, J < F".

121

15

Proof. ( m - 1 ) K x is S-big, so for all sufficiently big and divisible integers TO, the linear series \ r o ( m- 1 ) K x - 2g*L( is non-empty. Let D be a general member of this linear series. Replacying X by an appropriate birational model, we may assume that ImKx, I = IH"I F" where IH"J is base point free and F" has normal crossings support. Moreover we may assume that

+

I(ro(m- 1 ) K x - 2g*L)Ix,I = IN1+ G

148


+

where IN1 is base point free, G Fm has normal crossings support. Let D = C diDi with Di reduced irreducible divisors, and let d := cdi. Since s E S is a general point, also DilxS is reduced and so for G = CgiGi we also have gi 5 d. Let t > then

e,

m

G Fm+--J + F m tr0 Since K ( S )2 0, for r sufficiently big and divisible, the linear series g*l(t - l ) r ( m1 )K s I is non-empty. By [V], for all sufficiently big and divisible integers r > 0, there is a surjection (t- 1 ) T ( m - 1 )

H O ( Xwx/s

(t-l)T(m-l)

@g*L"/To)@C(X) --+ H o ( X s wx, ,

1.

Assume that rom divides r. Let B1, Bz, B3 be general divisors in the linear series Iw$;)'("-) @g*L'I'OI, IrO(m - 1)Kx - 2g*LI, g*l(t - l ) r ( m- 1)Ksl respectively. It is easy to see that

B1Ix,

= f i ( t - l ) ' ( m - l ) + F(t-1)'("-1) 4 fi(t-l)'("-l)

+

(t - l ) r ( m- 1 ) m

Frn3

is a general member of lH(t-l)'(m-l)I. where Consider now the map of linear series

r

I(t - l ) r ( m- l ) K X / s + -g*LJ x Iro(m - 1)Kx - 2g*LJX'/"O To

r Itr(m - 1 ) K x - -g*LJ. r0 We remark that B ~ ( x=, 0 and Bzlx, = N G where is a general member of the base point free linear series IN[. It is easy to see that xg*l(t - l ) r ( m- 1)Ksl

+ -

+

B1

1

+ $& f B3 tr

+ (t

tr

m

-

l ) r ( m- 1 ) F" trm

N ++ -1G tro tro

+ B3. Let u : X' X be a birational morphism such that J t r ( m- 1)Kx! - $v*g*LI = @ + ]MI where @ + K x t / x is a divisor with normal crossings support and /MI is base point free. We have v*B + t r ( m - l ) K X , / x E @ + IMI. X is smooth and { g l x , } is klt and hence Define B := B1+ $Bz

-+

l v *{

B

I& 11 4 Kx:/x,.

Since

ImKx:I = ImKx,I+mKx:/x, = f * l H m l + F m + m K x : / x , , = f *F" m K x ; / x , is the fixed divisor of ImKx; I.

f * IH"I is free and F'"

+


+

149

We remark that glxs = ( $ 1 ~ ~ )Lglx,], and for general s E S, u*(B(x,)= 1 For B’ a general member of IM 1, one has

(u* B) Xi.

B - L~*{,lX,)

+

B + u*l-lx,J + mKx:/x,

4 V*Frn+ mKx:/x,. tr So for a = tro, b = $-, and for a general divisor B’ E lab(m - 1)KXl - bu*g*LI, one sees that B’ has normal crossings support and L%lx:J 4 Pm.Replacyng B with B’ and X with X‘ the lemma follows. 0

- Kx:/x,l

Theorem 5.2. Let X be a smooth projective variety of general type and albx(X) c Alb(X) its Albanese image. Let F := X, be the fiber over a general z E albx(X). Let m l 2 mz L 2 be any integers such that (miKF(is non empty and ((ml+mz)KF( defines a birational map. Then I(ml 2mz)Kx + PI induces a birational map for any P E Pico(X).

+

Proof. We sketch the proof here. For more detail, please see [CHI. Let a := albx : X -+ A be the Albanese map, v : Z -+ albx(X) an appropriate desingulaxization. Fix H an ample line bundle on A. Let m = ml +mz. For r >> 0 sufficiently divisible, pick a divisor B E Ir(m1 - 1)(m2 - l ) ( m - 1)Kx - a*HI. Replacing X by an appropriate birational model, one may assume that albx factors through a morphism f : X -+ 2 and that B has normal crossings support and by Lemma 5.1 that for general z E 2 (after replacying H by an appropriate multiple) B Lr(m-lEmz-l)I Ixz 4 L r ~ ~ - ~ ~ ( m 1Ixz - 1 j4 I F~~ and Lr(ml-ll(mz-l JIIX, + Fm. We define ~

~

‘

7

Lz := (mz - 1)Kx -

B r ( m - l)(ml - 1)1

1

150


Therefore, since a*H = f * ( v * H ) ,L1, Lz and L3 are numerically equivalent to the pull back of a nef and big Q-divisor on 2 plus a klt divisor. In particular

hi(Z,f * ( w x @ L j ) @ ~ *=Phi(A, ) a , ( w ~ @ L j ) @ P=) 0 for all i > 0, j j = 1,2,3.

= 1 , 2 , and all

P

E Pico(A). Therefore,

a,(wx@Lj) is I T o for

By a general fiber, we mean a smooth fiber over z E Z such that X, satifies the assumption and

Ria*( W X @ L j )@k(z ) Hi(X, ,wx, @ (Lj) ) , for all z , j . By a general point y E X,, we mean a point in X , which is not in the base loci of IKx, Lj 1 for all j. The idea is to seperate general points y1, yz. For any such yi, yi is not an essential base point of IKx L j ( thanks to the local isomorphism and the choice that yi is not in the base loci of IKx, $1. It follows that yi is not a base point of l2Kx Li Lz PI for all P. By comparing the base loci, yi is not a base point of 1Kx L3 PI for all P. We then shows that a,(wx@Lg@Zyi)is ITo. Hence for general P E Pico(A), one can find a divisor in IKx L3 PI passing through y1 but not yz. We pick a divisor in IKx L1 - PI not passing through y1,yz. Thus one sees that )2Kx L3 L1 I seperates y1 and y ~ . Turning into a more detail discussion, we proceed the comparison of base loci. We consider the divisor

+

+

+

+ + + + +

+ +

+

+ + B

B

B

p : = 1r ( m 1 - l)(mz - 1)' - ' r ( m- l)(rnl - I)'

- 4 ( m - l)(rnz

-

1.

1)

Write B = biBi where the Bi are the distinct reduced irreducible components of B. Then from the inequalities

bi Ir ( m 1 - l)(mz - 1)'

it follows that

p

bi 4 ( m - l)(m1 - 1)

+ r(m

-

bi 1)(rn2 - 1)12

bi bi 4 ( m - l)(m1 - 1)' + 4 ( m - l)(m2 - 1)J is effective. Therefore, we have a map of linear series

IKx + L3I

3l2Kx + L1+ Lzl.

It is possible to arrange that [T(ml-$(m2-l)j 4 BslmKxI. Comparing base loci, one sees that

+ L 3 ) ) = hO(X,O x ( 2 K x + 2L)) = hO(X,U x ( m K x ) ) .

hO(X,Ux(Kx

Moreover, it is easy to see that


151

for all P E Pico(A). By [HP] Proposition 2.15, one also sees that h o ( X ,O x ( m K x ) )= h o ( X ,O x ( m K x ) @ P for ) all P E Pico(A). In particular, for all P E Pico(A) one has an isomorphism

I K x + M + P \ 31 2 K X + L l + L 2 + P I and therefore if y $! BslmlKx,I U ImzKx, 1, one has that y $! BslKx L3 PI. We claim that if yl,y2 E X are general points, then for general P E Pico(A) one has (PIIcx+L3+PI(Y1) # 'plKX+L3+PI(Y2). To see this, Let zi = a(yi). Assume that z = z1 = 22 (if z1 # z2 the assertion will follow in a similar but easier fashion). Since yt E X are general, we may a s u m e that z E 2 is also general. Let F = X,, since ( ~ 1 =~ ~ ~ 1 is birational, we may assume that the homomorphism H o ( F ,( w x @ ' L 3 ) 1 ~-)+ C(y1) @ C(y2) is surjective. Since z E 2 is general, we have

+ +

a* (wX@&3)@@( (. Z ) ) One sees that

f*( W X @ k ? ) @ c ( Z )

H o( F , (wx@L3)IF).

a*(wx@~3@Z,,)@@(v(z)) = = ker (a*(wX@L3)@@(V(Z))--tH0(F,( L J X @ & ) I F ) u*(WX@L3@2,,,y,)@@(V(Z))

-+

@(yl)) ,

=

leer (a*(wx@'L3)@.@(v(z))H o ( F ,( W X @ ' L S ) I F ) @(yl) @ @(y2)) . It follows that a*(wx@L3@1y1,y,) # a* (wX@L3@Zyl). Consider the exact sequence ----f

0

-

-

a*(wx@L3@Z,,,yz)

+

a*(wx@L3@Zg1) +0 .2 + 0.

Since h 0 ( a + ( ~ ~ @ L 3 @ Z y l= @ho(a,(wx@L3@P)) P)) - 1 for any P E Pico(A). Since hi(a,(wx@L3@P))= 0 for all i > 0, it follows that U , ( W X @ L ~ @satisfies Z ~ ~ ) the I.T. and hence by Proposition 2.3

hO(a,(wx@Ls@Z,,,,,)@Q) = ho(a*(wx@L3@Zy,)@Q) - 1 for general Q E Pico(A). Therefore for general Q E Pico(A)

ho([email protected]@Zyl,yz@Q) = h0(~x@L3@Zyl@ -Q 1)= ho(wx@'L3@Q) -2 It's then easy to see that the linear series I(m1 + 2 m 2 ) K x PI also induces birational morphisms. 0

+

Corollary 5.3. Let X be of general type, then i) zf dim(X) - u ( X ) 5 1 then )6KxI induces a birational morphism, ii) dim(X) - u(X) = 2 then 17KxI induces a birationnl morphism.

152


Proof. For curves of general type, 12KxI # 0 and 13Kx( defines a birational map. Thus it suffices to take ml = m2 = 2 as in the previous Theorem. And for surface of general type, ) 2 K x ) # 0 and )5KxI defines a birational map by [Bo]. Thus it 0 suffices to take rnl = 3,ma = 2.

When X is of maximal Albanese dimension, it is possible to improve the above result. Theorem 5.4. Let X be of general type and maximal Albanese dimension. If ~ ( w x >) 0 , then 13KxI is birational. Proof. By [EL2], since ~ ( w x >) 0, it follows that for all P E Pic0(X),ho(X,w x @ P ) > 0. There exists an open subset U c X such that for any x E U , and general Q E Pico(X) (depending on z), there is a section of Ho(X,w x @ Q ) not vanishing at x . Therefore] for any x E U and for any P E Pico(X), there is a section of H O (x,W ~ R Pnot ) vanishing at 2. Let X I , x2 be two general points in U . Proceeding as in the proof of the previous theorem, we see that for general Q E Pico(X),there is a section of w:@Q vanishing on x1 but not on 5 2 . For a fixed P E Pic0(X),we choose general Q E Pico(X) and then consider the map of linear series

+

+

13Kx PI. JKx- Q +PI x 12Kx QI We conclude that for any fixed P E Pico(X), there is a section of w$@P vanishing on x1 but not on 5 2 . 0 REFERENCES

[AS] U. Angehrn, Y. T . Siu, Effective freeness and point seperation f o r adjoint bundles, Invent. Math. 122 (1995)2, 291-308. [Bo] E. Bombieri, Canonical models of surfaces of general type, IHES Publ. Math. No. 42 (1973), 171-219. [CHI J. A. Chen, C. D. Hacon, Effective generation of adjoint linear series of irregular varieties, Preprint. [EL11 L. Ein, R. Lazarsfeld, Global generation of pluricanonical and adjoint linear series on smooth projective threefolds, Jour. AMS 6,No. 4 (1993),875-903. [EL21 L. Ein, R. Lazarsfeld, Singularities of theta divisors, and birational geometry of irregular varieties, Jour. AMS 10,No. 1 (1997),243-258. [Ha] R. Hartshorne, Residue and Duality, Lecture Notes in Mathematics 20. Springer-Verlag, 1966. [HP] C. D.Hacon and R. Pardini, O n the birational geometry of varieties of m m i m a l Albanese dimension To appear in Jour. fur die Reine Angew. Math. [Ka] Y . Kawamata, O n Fujita'sfreeness conjecture for 3-folds and 4-folds, Math. Ann. 308,(1997) 491-505. [KO] J. Kollbr, Shafarevich Maps and Automorphic Forms, Princeton University Press, 1995. [Mu] S. Mukai, Duality between D ( X ) and D(X) with its application to Picard sheaves, Nagoya Math. Jour. 81, (1981)153-175.


153

[Re] I. Reider, Vector bundles of rank 2 and linear systems o n algebraic surfaces, Ann. Math. 127, [V]

(1988) 221-233. E. Viehweg, W e a k positivity and the additivity of the Kodaira d i m e n s i o n for certain fiber spaces Algebraic Varieties and analytic Varieties (Tokyo 1981), 329-353, Adv. Stud. Pure Math. 1. North-Holland, Amsterdam, 1983.

Hecke curves on the moduli space of vector bundles over an algebraic curve Jun-Muk Hwang Throughout the paper, we will work over the complex numbers. In [Hw], we studied the geometry of the moduli space Mi of vector bundles of rank 2 with a fixed determinant of degree i over a smooth projective algebraic curve of genus g 2 2 via Hecke curves on it. We specifically obtained the following two results:

([Hw,Theorem 11) The tangent bundle of Mi is stable i f i is odd. ([Hw,Theorem 21) For g 2 5, the set of generically finite surjective morphisms from a fixed complete variety to Mi is countable for any i. In this paper, we give partial generalizations of these to the cases of moduli spaces of vector bundles of higher rank. More precisely, let M be the moduli space of semi-stable bundles of rank r with a fixed determinant and M S be the smooth locus of M . Then we have the following.

Corollary 1 The tangent bundle T ( M s ) is simple if g

2 4.

+

Corollary 3 If X is a general curve of genus g > 2r 1, the set of generically finite surjective morphisms from a fixed complete variety to M is countable. These corollaries follow from our main results, Theorem 3 and Theorem 4 stated in Section 3, concerning the variety of standard rational tangents on M . Each of the two theorems is of independent interest in its own right and perhaps more important than the corollaries above. It seems hard to strengthen Corollary 1 to the stability of T ( M " ) . In the rank 2 case of [Hw], we were able to exploit the simple structure of ruled surfaces. In order to generalize the same idea to higher rank cases, we need a good knowledge of (double) projective bundles of higher rank over curves. On the other hand, a refinement of our technique may improve Corollary 3 to include some cases with g 5 2r 1. For r = 2 , the proof of Corollary 3 in Section 3 gives another proof of [Hw,Theorem 21 for general curves of genus g > 5. Our proof of Corollary 3 below is rather elementary, while the proof in [Hw] uses deeper results about Mi. M.S. Narasimhan ([Na]) informed us that he had a different proof of the simplicity of T ( M ) ,which applies t o any g 2 2 (unpublished).

+

'Supported by Grant No. 98-0701-01-5-L from the KOSEF.

155

156

JUN-MUK HWANG

As in [Hw],our main tools are the theory of standard rational tangents developed in our joint work with N. Mok ([HM])and the geometry of Hecke curves on M . Hecke curves were introduced by Narasimhan and Ramanan and most of their properties needed here are found in “R2,Section 51. However, since the main concern in [NR2] was Hecke cycles, which are not curves, the properties of the Hecke curves were only implicitly studied there. For the readers’ convenience, we will rewrite some of the details in Sections 2 and 3 below with references to the relevant part of [NR2]. There are a number of interesting open questions about Hecke curves. We mention two of them, Question 1 and Question 2 below, which are most interesting from the viewpoint of the theory of standard rational tangents on Fano varieties. Acknowledgment We are grateful to Professor M.S. Narasimhan for valuable discussions and the referee for suggestions which improved the clarity of the paper.

1

Generalities on the variety of standard rational tangents

In this section, we recall some results from the theory of the variety of standard rational tangents ([HM]). Let M be a Fano variety of Picard number 1. We will assume that there exists a component H of the Hilbert scheme of complete curves on M such that (1) generic members of H lie in the smooth locus M S of M and sweep out an open subset of M S and that (2) the subscheme Hy c H consisting of members of H passing through a generic point y E M is an irreducible smooth complete variety of which every member is an irreducible smooth rational curve contained in M“. For a smooth point y E M , let T,(M) be the tangent space to M at y. Define the tangent morphism 7, : Hy

+

PT,(M)

by sending [C]E H, corresponding to a smooth rational curve C Ty([c]) :=

c M s to

PT,(C).

From The assumption (2) above, it follows that a generic member of H, has normal bundle of the form O(1)P @ Oq,p q = dim M ([Ko,IV.2.9]), implying that 7, is generically finite over its image. The image of T, will be denoted by C, and called the variety of standard rational tangents at the generic point y associated to the

+

HECKE CURVES ON THE MODULI SPACE OF VECTOR BUNDLES

157

family H . The dimension of C, is p, the number of O(1)-factors in the decomposition of the normal bundle of a generic member of H,. The following is a special case of [HM, Theorem 1.41 (Note the condition in [HM,Theorem 1.21 is satisfied when C, is smooth and irreducible by the remark after [HM,Theorem 1.21.). In [HM], it was assumed that M is smooth. But the proof works verbatim even when M has singularity under the assumptions (1) and (2) above. Let A denote a unit disc in C .

Theorem 1 In the above situation, assume that ry is an embedding. Then for any complete variety Z and a family of generically finite surjective morphisms {ft : Z + M , t E A}, there exists a family of biregular automorphisms {gt E A u t ( M ) , g o = id^, It1 < E } such that f t = gt 0 fo for all Jtl< E for some E > 0. In particular, if M has no vector fields, we have the following rigidity:

Theorem 1' I n Theorem I , assume furthermore that A u t ( M ) is 0-dimensional. Then any family of generically finite surjective morphisms { f t : Z + MIt E A} from a fixed complete variety Z is trivial, namely, f t = fo for all t E A. Let T ( M " ) be the tangent bundle of the smooth locus of M . Recall that a vector bundle is simple if the scalar multiplications are the only endomorphisms of the vector bundle.

Theorem 2 Let M be a Fano variety which has a component H of the Halbert scheme satisfying the conditions ( I ) and (2) above. Assume in addition that the variety of standard rational tangents C, at a generic point y E M is non-degenerate in P T , ( M ) . Then the tangent bundle T ( M S )is simple. Pro05 Let x be an endomorphism of T ( M " ) . Let y E M be a generic point and v E T,(M) be a vector tangent to a generic member C c M" of H,. Let V be a vector field on C extending v having two distinct zeroes. Then x(V) is a section of T ( M " ) l c vanishing at two distinct points where V vanishes. R o m the decomposition type of the normal bundle of C, x(V) must be tangent to C. Since V and x(V)have two distinct zeroes at the same points, they must be proportional. It follows that v is an eigenvector of x in T,(MS). Since this is true for any generic choice of v in C, which is non-degenerate in PT,(M"), we see that every vector of T , ( M S ) is an eigenvector of x. Thus x acts as a scalar multiplication on T , ( M ) for generic y E M . The eigenvalues must be constant along members of H because V and x(G) are just a constant multiple of each other. Since M has Picard number one, any point of M Sis joined to 2 by a connected chain of curves Ci E H ([HM, Section 31). Noting that the eigenvalue of the scalar transformation x is constant on Ci, we see that x is globally constant scalar multiplication on M " , meaning the simplicity of T ( M S ) . 0

158

2

JUN-MUK HWANG

Hecke curves

Let X be a smooth projective curve of genus g and let M denote the moduli space of semi-stable bundles over X of rank r and with a fixed determinant of degree d. It is well-known that M is a Fano variety of Picard number 1. The smooth locus M s of M corresponds to stable bundles on X. Let U ( r , d )be the moduli space of semi-stable bundles of rank r and degree d . There is a natural surjective morphism U ( r , d ) --+ J a c ( X ) whose fiber is isomorphic to M . The dimension of U ( r , d ) is r 2 ( g - 1) 1 and the dimension of M is ( T ~- l ) ( g - 1).

+

Given two nonnegative integers k and 1, a vector bundle W of rank r and degree d is (k, 1)-stable, if, for every proper subbundle V of W, we have

The usual stability is equivalent to (0, 0)-stability. A ( k ,1)-stable bundle is ( k ,1 - 1)stable for 1 > 0. The dual bundle of a (k, 1)-stable bundle is (1, k)-stable. We need the next two Propositions proved in [NRZ] to explain the definition of Hecke curves.

Proposition 1 ([NR2,5.4]) If g 2 4, a generic point [W]E M corresponds to a (1,l)-stable bundle W.

Ox+ 0 , Proposition 2 ([NR2,5.5]) Given an exact sequence 0 + V + W where V,W are vector bundles on X and Oxis the skyscraper sheaf at a point x E X , if W is ( k ,1)-stable, then V is (k, 1 - 1)-stable. --+

Throughout this section, we assume that g 1 4 so that a generic point [W] E M corresponds to a (1,1)-stable bundle W over X by Proposition 1. Denote by W* the dual bundle and P W the projectivization consisting of lines through the origin on each fiber. For x E X and C E PW:, consider a new vector bundle Wc defined by

0

--f

wc

--+

w

--+

(W,/C') 8 Ox-----t 0

where denotes the hyperplane in W, annihilated by C. Let L : W i + W, be the homomorphism between the fibers at x induced by the sheaf map Wc + W. The kernel of L , Ker(L), is a 1-dimensional subspace of the fiber W i and its annihilator (Ker(L))' is a hyperplane in (Wc):. Let 1 be a line in P W i containing the point [ K e r ( ~ ) For ] . each point [l] E 1 corresponding to a 1-dimensional subspace 1 c W i , consider the vector bundle El defined by

0

--+

EL

--+

(WC)* + [(wc);/l*] 8 o x

--+

0


where I-'- c (Wc): is the hyperplane annihilating 1. This vector bundle for each [l] E 1 by Proposition 2. It is easy to check that for 1 = Ker(L),

159 is stable

-

W W L ) 2 W*. It follows that { ( $ ) * ; l E 1) defines a rational curve passing through [W]in M". A rational curve on M" constructed this way is called a Hecke curve. Using [NR2,5.9], one can show that a Hecke curve is smooth. In view of [NR2,5.16], it is easy to check that a Hecke curve has degree 2r with respect to KG1. The following question seems to be of great interest. Question 1 Is a Hecke curve a rational curve of minimal degree passing through a generic point of M? Conversely, is a rational curve of minimal degree passing through a generic point of M necessarily a Hecke curve? By [DN], K i l = 2nB where 0 is the generator of Pic(M) and n = ( r , d ) . Thus for d = 0, a Hecke curve has degree 1 with respect to 0 and the first part of Question 1 is affirmative in this case, as well as in case r = 2 ([Hw,Proposition 81).

-

Let IT : PW* X be the natural projection. On P W * , consider the relative cotangent bundle R* of the fibration IT. The projective bundle PO2"over PW* is a smooth projective variety of dimension 2r -2. The set of all lines in P W i containing the point [ K e r ( ~ )is] naturally isomorphic to P(Wi/Ker(L)) Z Pa;. The argument of [NR2,5.13]shows that Hecke curves associated to two distinct points of PO" are distinct rational curves on M . Thus PR" is naturally isomorphic to the variety of all Hecke curves through [W].

Proposition 3 Hecke curves are dense in an irreducible component of the Hilbert scheme of curves on M . Proof. Since dimPR" = 2r - 2, the set of all Hecke curves have dimension dim(M)+2r-3. A generic Hecke curve must have semi-positive normal bundle. Thus the space of all deformations of a Hecke curve C has dimension dim(M)+C.KG1 -3. Since C . K i l = 2r, all small deformations of a Hecke curve are again Hecke curves. 0

It follows that the component H of the Hilbert scheme of M corresponding to Hecke curves satisfies the conditions (1) and (2) for H in Section 1.

3

Tangent morphism for Hecke curves

The tangent morphism T[WI : Hiw] = P V + PTlw]( M ) for a (1,l)-stable bundle W on X is defined by associating to 77 E PR" the tangent vector of the corresponding

160

JUN-MUK HWANG

Hecke curve at [W],namely, 7 [ W ] ( r ] ) :=

pqw](c,,)

where C,, c M is the Hecke curve defined by

r].

Theorem 3 The tangent morphism ~ [ W for I a (1,l)-stable bundle W is the morphism defined b y the complete linear system of the line bundle

on Pa2"where 11, is the composition of the two projections

P02" A P W * A

x.

and E is the dual tautological line bundle of the projectivization satisfying cp& = T", T" denoting the relative tangent bundle of IT. Proof. The proof is essentially the same as that of [Hw,Proposition 111, once we note that Ho(H[wl,Lw)* is canonically isomorphic to Tlwl(A4). Let us just check this fact. HO(PR", Lw) = HO(PW*,cp*E @ IT*Kx) =

Ho(PW*,T" @ n*Kx)

= H o ( X ,x,TX@ Kx) = H o ( X ,adw @ Kx)

where adw is the bundle of traceless endomorphisms of W on X . By Serre duality, the last line is canonically dual to H 1 ( X , a d ( W ) )which is canonically isomorphic T p q ( M ) by standard deformation theory of bundles. 0 From Theorem 3, the variety of standard rational tangents C[WI = ~ [ w( H ] f w ] is) non-degenerate in PT[w]( M ) . By Theorem 2, we get

Corollary 1 The tangent bundle of M" is simple if g 2 4. By the result of Kebekus ([Ke,Theorem 3.4]), the tangent morphism is finite over its image, whence follows

Corollary 2 The line bundle vector bundle W on X .

E @ $*Kx on PO" is ample for any ( 1 ,1)-stable

When T = 2, Corollary 2 is well-known in terms of geometry of ruled surfaces (cf. [Hw]). It is probably possible to give a more direct proof also for higher ranks.


161

In regard to Theorem 1, one of the most intriguing questions about the tangent morphism is the following:

Question 2 Given a suficiently generic stable bundle W on X, is the lane bundle L w very ample on Pa2" ? We have the following partial answer.

Theorem 4 If X is a general curve of genus g > 2r + 1, the tangent morphism T[W]: H[wl --t P q w l ( M ) is an embedding for a generic point [W] E M . For the proof, we need two lemmas. The first one is an honest generalization of [NR2,Proposition 5.41.

Lemma 1 If g > C

+ 1, then a generic point of M

is (0, [)-stable.

Proof. As in [NR2,Proposition 5.41, we count the dimension of the space of non-(0,C)-stable bundles. Recall that a vector bundle W of rank T and degree d is (0,C)-stable, if the rank q and the degree 6 of any proper subbundle satisfies qC < dq - 6r. Suppose a stable bundle W is not (0,C)-stable and V c W be a subbundle of rank q and degree 6 with q l 2 dq - 6r. Since non-stable bundles can be deformed to stable bundles ([NRl,Proposition 2.6]), we may assume that V and W/V are stable in dimension counting. The dimension of deformations of V is equal to dimUx(q,6) = q2(g - 1)

+1

while W/V has deformation of dimension dimUx(r - q,d - S) = (r - q ) 2 ( g - 1)

+ 1.

The dimension of extensions of W/V by V is =_

hl(X, H m ( W / V , V)) = -deg(Hom(W/V, V)) = -q(6

-

d ) - (r - q)6

+ (g - 1) . rk(Hom(W/V, V)) + ho(X,Hom(W/V, V))

+ (g - l)q(r - q )

because h o ( X ,Hom(W/V, V ) )= 0 from the stability of W. Thus the dimension of non-trivial extensions is qd - r6 (g - l)q(r - q ) - 1. It follows that the dimension of the space of non-(0, C)-stable bundles is at most

+

- 1)

+ 11 +

[(T

- d 2 ( g - 1)+ 11 + [4d - rd + (9 - 1Mr - Q ) - 11

= (g - 1)(q2

+ r2 - rq) + qd - r6 + 1.

162

JUN-MUK HWANG

The right hand side is strictly smaller than dimUx(r,d) = r2(g - 1)+ 1 if (g - l)(rq - q2) > qd - rb. But from the assumption g - 1 > e and the hypothesis qi! 2 qd - r b on q and b, (g - l)q(r - q )

> i!q 2 qd - rb.

0

The next lemma is a generalization of "R2,Lemma 5.61.

+

Lemma 2 Let r be a positive integer. Assume that ho(X,O(rx ry)) = 1 for any x,y E X . Then for any (0,2r)-stable bundle W of rank r and any stable bundle W' o r rank r with det(W') = det(W) @ 0 ( - r x - ry), we have (a) if f : W' W is a non-zero homomorphism, f is of maximal rank, (ii) ho(X,Hom(W', W)) 5 1. --f

Proof. If f is not of maximal rank, let W'

----f

V'

1

-

0

W - V - Q be the factorization off with V Im(f ) and V' 2 W'/Ker( f ) so that q := rk(V) = rk(V') < r and V' -+ V is of maximal rank, implying deg(V) 2 deg(V'). Since W' is stable, d4V) 4

2- deg(V') > -deg(W') r

4

- deg(W) - 2 r -

r

which contradicts the (0,2r)-stability of W. This proves (i). Suppose f and g are two linearly independent homomorphisms W' -+ W. Choose z # x, y. Then h = af bg # 0 for some a, b E C must induce a singular element h, of Hom(WL, W,). By (i) h has maximal rank. Let

+

ATh : ATW' E det(W) @ 0 ( - r x - ry)

+

-

A'W = det(W)

be the determinant of h. Since h o ( X ,O(rx ry)) = 1, R'h has zero precisely at r x + r y . This is a contradiction to the fact that h, is singular. This finishes the proof of (ii). 0 Proof of Theorem 4. Write L for Lw for simplicity. For any x 6 X , the line bundle L restricted to the fiber y!-'(x) = P(Cll(PW:)) is very ample. Thus L is very ample on PO" if for any x, y E X, the case x = y included, the restriction map HO(Pfl",L )

-

H0W1(X

is surjective. From the exact sequence

+ Y), LI+-'(s+y))

HECKE CURVES ON THE MODULI SPACE O F VECTOR BUNDLES

0 + L 8+*O(-IC

-

y)

--f

L

+

Ll$-l(z+y)

-

163

0,

the surjectivity is guaranteed if

H'(PW, L 8+*o(-Ic - y)) = H 1 ( X , a d w 8 K ~ ( - I Cy ) )

+ +

or its dual H o ( X ,adw 8 O(z y)) vanishes. By Lemma 1, we may assume that W is (0,2r)-stable from g > 2 r 1. If X is a general curve of genus g > 2 r 1, we see that the assumption of Lemma 2 is satisfied, in view of the dimension count of curves which have meromorphic functions with poles of the type TIC r y as in [ACGH,p.213]. By Riemann-Hurwitz formula, there will be at most (29 - 2) 47- - 2 ( r - 1) = 2g 2r branch points of the meromorphic function X PI in addition to the pole. Thus up to the automorphism of P I preserving 03, there are at most 29 2r - 2 branch points. It follows that if 39 - 3 > 29 2 r - 2, namely, g > 2r 4-1, then a generic genus g curve cannot have such a meromorphic function. Setting W' = W 8 0 ( - 5 - y) in Lemma 2 , we get

+

+

+

+

--f

+

+

1 2

hO(X,Hom(W',W))

= h O ( X End(W) , 8 O(IC =

~ O ( X ,O(Z

meaning that H o ( X ,adw 8 O ( x

+ y)) +

+ y))

~ O ( X ,adw

8 O(X

+ 2/11

+ y)) = 0, as desired. 0

Noting that A u t ( M ) is 0-dimensional by [NRl], we apply Theorem 1' to get

+

Corollary 3 FOT a general curve X of genus g > 2r 1 and any complete variety Z of dimension equal to dim M , the set of dominant morphisms from Z to M is countable. REFERENCES [ACGH] Arbarello, E., Cornalba, M., Griffiths, P. and Harris, J.: Geometry of algebraic curues, Volume I. Grundlehren der math. Wissenschaften 267, Springer Verlag, 1985 [DN] Drezet, J.-M. and Narasimhan, M. S.: Groupe de Picard des variktes de modules de fibres semi-stables sur les courbes algkbriques. Invent. math. 97 (1989) 53-94 [HM] Hwang, J.-M. and Mok, N.: Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1. J. Math. Pures Appl. 80 (2001) 563-575 [Hw] Hwang, J.-M.: Tangent vectors t o Hecke curves on the moduli space of rank 2 bundles over an algebraic curve. Duke Math. Journal 101 (2000) 179-187 [Ke] Kebekus, S.: Families of singular rational curves. J. Alg. Geom. 11 (2002) 245-256

164

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[KO]Ko110, J.: Rational curves o n algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, Band 32, Springer Verlag, 1996 "a] Narasimhan, M. S.: private communication.

[NRl] Narasimhan, M.S. and Ramanan, S.: Deformations of the moduli space of vector bundles over an algebraic curve. Ann. Math. 101 (1975) 391-417 "R2) Narasimhan, M.S. and Ramanan, S.: Geometry of Hecke cycles I. in C. P. Ramanujam-a tribute. Springer Verlag, 1978, p.291-345

Korea Institute for Advanced Study 207-43 Cheongryangri-dong Seoul, 130-012, Korea [email protected]. kr

MINIMAL RESOLUTION VIA GROBNER BASIS YUKARI IT0 In memory of Meeyoug Kim

ABSTRACT.The minimal resolution of cyclic quotient singularity is well known and the McKay correspondence can be defined as a correspondence between the exceptional curves of the minimal resolution and the special representations. In this paper, we give a construction of the minimal resolution and explain the generalized McKay correspondence with the special representations in terms of the Grobner basis.

CONTENTS 1. Introduction 2. G-Hilbert scheme 3. Proof of Theorem 1.1 4. Application to the generalized McKay correspondence 5. Example references

165 166 168 171 172 174

1. INTRODUCTION

Let G be a finite small group in GL(2,C) and X the quotient C2/G. In this case, it is known that there exists the minimal resolution of the quotient singularity, where a group is small if it acts freely on C? \ (0). The history of the minimal resolution is very long and many algebraic geometers have contributed to the construction of the resolution. To reach the resolution, several tools have been established such as blowing-up, Newton diagram, continued fraction and toric resolution. 1991 Mathematics Subject Classafication. 14C05, 14E15. Key words and phrases. quotient singularity, minimal resolution, special representation, Grobner basis. The author is partially supported by JSPS, the Grant-in-aid for Scientific Research (No.13740019).

165

166

YUKARI IT0

In this paper, we would like to introduce a new construction of the minimal resolution via the Grobner basis. It is deeply related with the theory of the GHilbert schemes and based on the author's recent work [5]. The G-Hilbert scheme was introduced in a paper ( [ 6 ] )by Nakamura and the author and is isomorphic to the minirnal resolution of the quotient singularity. We will obtain the G-Hilbert scheme in terms of the Grobner basis as follows:

Theorem 1.1. Let G be a finite small cyclic group G in GL(2,C), and IG be an ideal of 0 ~ defined 2 by the free G-orbit. Then the Grobner f a n for the G-homogenous ideal IG determines a toric variety which is isomorphic to the minimal resolution of @/G. As an application of this theorem, we can explain the generalized McKay correspondence via the Grobner basis. The generalized McKay correspondence gives a bijection between the set of the exceptional curves of the minimal resolution and the set of the special irreducible representations of G. The special representation was defined by Wunram and Riemenschneider in terms of vector bundles on the minimal resolution, and the difference between special representation and non-special representation can be understood by the following theorem:

Theorem 1.2. There is a bijection between an irreducible special representation P k of G and a binomial generator in the initial ideal i n w ( I ) . By this theorem, we can calculate all the irreducible special representations in terms of the Grobner basis without any knowledge of vector bundles. We will show these two theorems in the following sections. This paper is organized as follows: First, we recall the results on the G-Hilbert scheme in the following section, and give some basic definitions in the theory of the the Grobner basis and the proof of Theorem 1.1in the next section. Then we apply the result to the generalized McKay correspondence with the special representations in section 4. In the the final section, we give an example with a picture of the Grobner fan.

2 . G-HILBERTSCHEME In this section, we discuss the G-Hilbert scheme in dimension 2. Let G be a finite small subgroup of G L ( 2 ,C) and X the quotient C2/G.

Definition 2.1. A G-Cluster Z is a G-invariant subscheme Z c C2 for which H o ( Z ,0 ~ is the ) regular representation of G, and the G-Halbert scheme is the moduli space of G-clusters. Remark 2.2.The G-Hilbert scheme HilbG(C2)is a roughly 2-dimensional component of G-fixed part of Hilbn(C2), where n = IG), and can be written as HilbG(C2)= { I E 0 ~ ~ :1G-invariant 1 ideal, &n/I

C[G]}.

MINIMAL RESOLUTION

167

VIA GROBNER BASIS

We have the following 2-dimensional results on the G-Hilbert scheme. Theorem 2.3. For any finite small subgroup G of GL(2,C), the G-Halbert scheme is the minimal resolution of C2/G. It was proved by

i

Ito-Nakamura([6]): Kidoh( [7]): Ishii( [4]):

G c SL(2, C), G is a cyclic subgroup of GL(2, C), G is a small subgroup of GL(2,C).

In the proof of the first case, they used the fact that the Hilbert scheme of n points on C2 Hilbn(C2) is a minimal resolution of the n-th symmetric products of C2. Under the action of G c SL(2,C), the smoothness of the fixed part and symplectic form are preserved, and then the G-Hilbert scheme, the 2-dimensional irreducible component of the G-fixed part of Hilbn(C2) is the minimal resolution of the quotient singularity C2/G. In the second case, we can show that the G-Hilbert scheme is a toric variety and discuss the stratification in terms of Hirzebruch-Jung continued fraction. The third case contains the above two cases, and the connectedness of the GHilbert schemes was proved in terms of derived category as an analogy of the result in dimension three by Bridgeland-King-Reid ( [ l ] ) . Let G be the cyclic group CT,a,generated by the matrix

(i

€0

where cT = 1

and gcd(r, a ) = 1, and consider the character map C [ x ,y ] C [ t ] / t Tgiven by x H t and y H ta. Then we have a corresponding character for each monomial in C [ x ,y ] . Let us recall the following theorem for the cyclic singularities. Here we can write down the G-Hilbert scheme as a set of ideals. This result gives the correspondence in Theorem 1.2. --f

Theorem 2.4 ([5]Theorem 3.9). For every finite cyclic groups in GL(2,C) and every G-cluster 2, the generator of the ideal I z can be chosen as the system of 3 equations: xa = ayc, yb = p x d , xa-dyb-c = ap,

{

where a and p are complex numbers and both xa and yc (resp. y b and x d ) correspond t o the same representation (or character). The pair (a,p) is a local affine coordinate near the point p and it is also obtained from the calculation with toric geometry. Moreover, each axis of the affine chart is just a exceptional curve or the original axis of C2. The exceptional curve is isomorphic to a P1and the points on it are written in a ratio sych as ( x a : y b ) (resp.

168

YUKARI IT0

(xd : y")) which corresponds to a special representation pa (resp. p d ) . The fixed point p is the intersection point of two exceptional curves Ea and Ed. Thus we can obtain the whole space of exceptional locus by deforming the point p and patching the afine pieces. Remark 2.5. We will define the special representations in section 4: it is enough to undestand that there exists a set of the irreducible representations of G for which each corresponds to an exceptional divisor. The relation between them in terms of the G-Hilbert scheme has already been discussed in the author's previous paper [5]. In the generalization to the higher dimensional cases, we have the following result, which was shown by Nakamura for abelian groups and Bridgeland-King-Reid for general cases:

Theorem 2.6 (Nakamura[8], Bridgeland-King-Reid[l]).For any finite subgroup G of SL(3,CC), the G-Hilbert scheme is a crepant resolution of C3/G. Here we use a similar defintion for 3-dimensional G-Hilbert schemes and a crepant resolution is a minimal resolution with trivial canonical bundle. Moreover, few examples of the G-Hilbert schemes are known in the higher dimension, and there are many problems yet to be proved, for instance, when the G-Hilbert scheme will be a smooth or crepant resolution.

3. PROOF OF THEOREM 1.1 Let us recall the basic notations for the Grobner basis and define the Grobner fan before the proof of Theorem 1.1. The author's notations are based on a book by Sturmfels [12]. Let k [ z l , . . .,x,] be a polynomial ring, and the monomials be denoted xa = xa1 $a2 . . . x2. 1 2 Fix a weight vector w = (w1, . . . ,w,) E Rn.

Definition 3.1. Let w 2 0 and < an arbitrary term order. Then the weight t e r n order 5wy 2 0,in(1)= (2, y7), c2[w]: 5wy > w, > ;wy 2 0, in(1)= (52,y5), > w, > gwy 2 0, in(1)= (23, y3), Q [ W ] : ijwy 3 cq[w] : w, > gwy 2 0, i n ( I )= (27, y). Thus we can obtain the minimal resolution via the Grobner basis. It can be oveserved that the same monomials appear in the the defining equations of the Gclusters and the initial ideals above. Moreover, we have the corresponding irreducible representations for each exceptional curve which is defined as an intersection of 2 affine pieces. of the closure of ~ [ w ] s .Let There is a line which is defined as the intersection Ui be an &ne piece deterined by the cone ci[w].For an exceptional curve El2 defined in Ul n U2, we have a non-monomial generator x - y5 in the initial ideal corresponding to a character E or an irreducible representation p1. Similarly, we obtain the other two non-monomial generators of the initial ideal and the corresponding representations:

174

YUKARI I T 0

~ 2 : 3 X’

E34:

-y3

x3 - y

tf H

p2, p3.

Thus we can obtain the special representations by Theorem 1.2. After writing up this paper, the author met D. Maclagan and she mentioned her similar result for toric Hilbert schemes.[9] REFERENCES

[l] T. Bridgeland, A. King and M. Reid, The McKay correspondence as a n equivalence of derived categories, 3. AMS, 14, (2001), 535-554. [2] H. Esnault, Reflexive modules on quotient surface singularities, J. Reine Angew. Math. 362 (1985), 63-71. (31 G. Gonzalez-Sprinberg and J.L. Verdier, Constwction ge‘ometrique de la correspondance de McKay, Ann. Sci. Ecole Norm. Sup. 16 (1983), 409-449. [4] A. Ishii, O n the McKay correspondence f o r a finite small subgroup of GL(2, C ) , t o appear in J. Reine Angew. Math. [5] Y. Ito, Special McKay correspondence, Actes de 1’Qcoled’QtQ“GQomQtriedes variQt6s toriques” (Grenoble, 2000), Skminaires et Congrlts, 6 (2002), (SociQtQMathQmatique de France), 213225. [6] Y. Ito and I. Nakamura, Hilbert schemes and simple singularities, in: New trends in Algebraic Geometry (Warwick, June 1996), K. Hulek and others Eds., CUP (1999), 151-233. (71 R. Kidoh, Hilbert schemes and cyclic quotient singularities, Hokkaido Mathematical Journal, 30 (2001), 91-103. [8] I. Nakarnura, Hilbert schemes of abelian group orbits, J. Algebraic Geom., 10, (2001), 757-779. [9] D. Maclagan and R. R. Thomas, The toric Hilbert scheme of a rank two lattice is smooth and irreducible, preprint, math.AG/0208031. [lo] J. McKay, Graphs, singularities and finite groups, Proc. Symp. Pure Math., 37 (1980), Amer. Math. SOC.183-186. [11] 0. Riemenschneider, Characterization and application of special reflexive modules on rational surface singularities, Institut Mittag-Leffler Report No.3 (1987). [12] B. Sturmfels, Grobner Bases and Convex Polytopes Univ. Lect. Series, 8, AMS (1995). [13] J. Wunram, Refledve modules on quotient surface singularities, Math Ann. 279 (1988), 583598. DEPARTMENT OF MATHEMATICS, TOKYO METROPOLITAN UNIVEFGITY, HACHIOJI,TOKYO 1920397, JAPAN E-mail address: yukariQcomp .metro-u.ac.jp

Deformation theory of smoothable semi log canonical surfaces Yongnam Lee abstract. The aim of the paper is to study the Q-Gorenstein deformation theory of log surfaces under suitable conditions. As its application, we consider compactifications of the space of double cover of P2 branched over curves of degree 4 or 6. Introduction Via the geometric invariant theory, Gieseker [Gi] proved the existence of a x smooth projective surfaces of genquasiprojective coarse moduli space M K ~ , for eral type with fixed numerical invariants K 2 and x. With the proof of bounds for log surfaces with given K 2 , Alexeev [A] clarified the construction of projective coarse moduli space of surfaces of general type with fixed K 2 that was started in [K-SB]. The compactified moduli space should include (possibly reducible) surfaces with ordinary double curves and certain other mild singularities. These surfaces are called smoothable stable surfaces. This notion of a smoothable stable surface can be generalized to a smoothable stable log surface similar as the generalization of a stable curve to a stable pointed curve. As applications of stable log surfaces, Hassett [Hasl] [Has21 considered a compactification Pj of plane curves of degree d ( d 2 4) by using the stable log surfaces and the Q-Gorenstein deformation theory of stable log surfaces. Then he succeeded to prove that Pi is isomorphic to Deligne-Mumford compactification of moduli space of curves of genus 3. He considered all possible plane curve singularities on the boundary of Deligne-Mumford compactification, and then he constructed corresponding stable log surfaces by using local stable reduction theorem. But it is already too complicated to manage if d 2 5 . Hacking [Hac] considered instead the family of compactifications given by moduli space P; of log surfaces ( X ,B ) such that Kx +aB is semi log canonical singularities and ample where $ < a 5 1. The compactification is simpler for lower a because it is allowed worse singularities for lower a. He gave a compactification Pd of plane curves of degree d by allowable family of stable pairs of degree d. Then he proved that Pd is a separated proper Deligne-Mumford stack and Pd is smooth if 3 f d. 'The research was supported by the Scientist Exchange Program by JSPS and KOSEF, and by the grant 1999-2-102-002-3 from the interdisciplinary Research program of the KOSEF. 2000 AMS Subject Classzfication. 14J10, 14J17.

175

176

YONGNAM LEE

The paper goes as follows : Section 1provides several notions of singularities as preliminaries. Section 2 presents the Q-Gorenstein deformation theory of pairs of log surfaces (X, B ) under the following assumptions : Assume that (X, B ) is a log surface such that 1. X is a semi log canonical surface, 2. B is a reduced Weil divisor on X, 3. B is Cartier on the index one cover of X. The third condition is necessary to compare between the Q-Gorenstein deformation of X and the Q-Gorenstein deformation of pairs (X, B ) . In paricular, the notion of stable pairs of degree d satisfies our assumtion. Our intention is to generalize the results in [L].

Theorem. Assume that X is a locally smoothable semi log terminal surface. Then Hoe;) = dPCP + H O ( O D ; ( C4))

c

c

,€Sing( X,D)

Di

9

where Sing(X,D ) is the set of singular points outside double curves D , singular point p is the type &(l, dpar - 1) with ( a ,r ) = 1 [K-SB], Di is a connected component of double curves D , and a singular point q an Di is not locally Q-factorial point. Therefore we obtain the infinitesimal smoothable deformations explicitly. Then the existence of smoothing can be interpreted by the relation between the tangent bundle of its minimal resolution and the normal bundle of the exceptional divisor. It holds the following:

Theorem. Let X be a locally smoothable semi log canonical surface and let p E Sing(X, D ) . Assume that p is permissible (dp = 1 and it admits a semistable resolution without changing a base [Ka]). Let ( S ,E ) be the pair of the minimal resolution of the germ (X,p) and the exceptional divisor in S . If H 2 ( 7 s ( - E ) )= 0 then Ho(7;,,) = C , 3 H 2 ( 7 j ) . In Section 3, as applications we consider compactifications of the space of double cover of P2 branched over curves of degree 4 or 6. If (X,B) be a stable pair of degree 4 or 6, then (X, 4B) has semi log canonical singularities. Let Y be a double cover of X branched over B. Then Y has also semi log canonical singularities. By using the Q-Gorenstein deformation theory presented in Section 2, we construct compactifications R4 and R6 of the space of double covering surfaces directly over P4 and P6.

Theorem. There are compactifications of the space of double cover of P2 branched over curves of degree 4 or 6.

DEFORMATION THEORY O F SMOOTHABLE SEMI LOG CANONICAL SURFACES 177

In [KL], we proved that if D is a plane curve of degree d and (P2,D)is a stable pair of degree d then D is a stable plane curve. The converse is also true under a suitable condition. These results make us possible to compare between the compactifications Pd and the compactifications of plane curves by Geometric Invariant Theory. In particular, we obtain the same classification of P4 as in [Hac] by construction of the stable pairs of degree 4 corresponding the semistable plane curves of degree 4. In a future work, we intend to classify the cases in P6 and intend to compare to this compactification R6 with Shah’s compactification [S] by using Geometric Invariant Theory of plane curves of degree 6. We work throughout over the complex number field C . The notation here follows Hartshorne’s Algebraic Geometry. 1. Preliminaries

The notion of discrepancy is the fundamental measure of the singularities of (X, D ) (cf. [K et] or [KM]).

Definition. Let X be a normal variety and D = C d i D i an effective Q-divisor such that Kx D is Q-Cartier. Let f : Y --+ X be a proper birational morphism. Then we can write

+

Ky

+ fcl(D)

f * ( K x+ D ) +

C a(E,D ) E

where fgl(D) is the proper transform of D , the sum runs over distinct prime divisors E c Y , and a(E,D ) E Q . This a(E,D ) is called the discrepancy of E with respect to (X, D ) ; it only depends on the divisor E, and not on the partial resolution Y. We define discrep(X, D ) = infE{a(E, D)IE is an exceptional divisor of X}. We say that (X, D ) , or K X + D is terminal canonical purely log terminal log canonical

if

discrep(X, D )

{

2 1 , 2-1.

Moreover, (X,D ) is Kawamata log terminal (klt) if (X, D ) is purely log terminal and di < 1 for every i; and (X, D ) is divisorial log terminal (dlt) if there exists a log resolution such that the exceptional locus consists of divisors with all a(E,D ) > -1. And we say that a log surface (X, D ) has semi log canonical (resp. semi log terminal) singularities if

178

YONGNAM LEE

1. X satisfies Serre's condition 5'2, 2. X has normal crossing singularities in codimension one, 3. K X D is Q-Cartier, and for any birational morphism f : Y semi-smooth surface Y we have

+

+

X from a

where all ai 2 -1 (resp. ai > -1). A surface Y is called semi-smooth if every closed point of Y is either smooth or normal crossing or a pinch point. Definition. A stable log surface is the pair ( X ,B ) where 1. X is a connected projective surface and B a reduced Weil divisor on X I 2. (condition on singularities) the pair ( X ,B ) has semi log canonical singularities, 3. (numerical condition) K X B is ample.

+

One can replace a general relatively log minimal model of a deformation of a smoothable stable log surface ( X ,B) + A with another relatively log minimal model, which has better singularities via base change and simultaneous resolution [Ka]. The notion of permissible singularities is due to Kawamata [Ka]. Definition. Let 7r : ( X , B ) + A be a deformation of a smoothable stable log surface ( X ,B ) . We say that 7r is a permissible degeneration if the following conditions are satisfied: The analytic germ of each point p E (XIB ) is isomorphic to one of the following germs, where 7r = t : 1. ( z y z = t ) , 2. ( z y = t ) c A , A = + ( a , r- a , 1,0) for ( a , r ) = 1, 3. ( z y = zT t") c A for some n.

+

In a permissible degeneration of ( X , B ) , the central fiber has smooth double curves, and each component of ( X , B ) has cyclic quotient singularities of the following two types: +(1,a) on the double curves and B , and $ ( a , r - a ) outside the double curves and B. And ( X , B ) admits a semistable model by the succession of weighted blowing up of singular points. This construction is explicitly given in [C]. We say that a pair (X, B ) has a permissible singularity at p if local germ at p has a permissible degeneration. 2. Q-Gorenstein deformation t h e o r y of pairs

Let X / d be a flat family of semi log canonical surfaces over a noetherian scheme. X / d is called Q-Gorenstein if w ! , ~ commutes with base change for all i E Z. And X / d is called weakly Q-Gorenstein if W X I A is Q-Cartier. In general, Q-Gorenstein

DEFORMATION THEORY OF SMOOTHABLE SEMI LOG CANONICAL SURFACES 179

implies weakly Q-Gorenstein. Hacking [Hac] showed that weakly Q-Gorenstein is the same as Q-Gorenstein if A is a discrete valuation ring. Let X be a semi log = Extk(Rx,Ox) for canonical surface. Consider the sheaves of deformation i = 0, 1,2. Define the subsheaves Ti of T i ,that represent the infinitesimal QGorenstein deformations. These subsheaves T i can be constructed via local index one cover. Let U c X be an analytic neighborhood with an index one cover U'. For the case of the field C , this index one cover is unique up to isomorphism. Define fi(U)as the invariant part of 7;,. And we define Ti by hypercohomology. It is obvious from the definition that c and ?j = 7;. Then T i = T$, and as the elements of T i mapped to H o ( T i ) . And the obstruction map can be

T$ 74

computed by T i c T i 2 T$. We consider the Q-Gorenstein deformation theory under the following assumption :

Assumption. Let (X, B) be a log surface such that 1. X is a semi log canonical surface, 2. B is a reduced Weil divisor on X , 3. B is Cartier on the index one cover of X . Let X be a semi log canonical surface with B a reduced Weil divisor on X. By the formalism of deformation of pairs [R] we have the long exact sequence 0 -+ T i , B -+ Tg @ Tg -+ Homx(R,y, O B ) T i , B -+ Tb @ T i -+ E x t i ( R x , 0,) -+ T$,B -+ T$ . The short exact sequence 0 -+ 0 x ( - B ) -+ O x -+ 08 -+ 0, where 0 x ( - B ) denotes the ideal sheaf of B, induces the long exact sequence: 0 ---t T$(-B) -+ T$ + HomX(Rx,,OB) -+ T i ( - B ) -+ T i Exti(ax,OB) -+ T$(-B) -+ T$ where T$ = Extk(f&,Ox), Tk(-B) = Ext$(S&,Ox(-B)) for i = 0,1,2. By these two long exact sequences, we obtain a long exact sequence for deformation of pairs, which is a generalization of the long exact sequence in a smooth case (cf. [Hasl]) : ---f

-+

0

T i ( - B ) -+ T$,B T ! T i ( - B ) T i , B-+ T B -+ T$(-B) T;.~. -+

-+

-+

-+ -+

( - B ) of 7;( - B ) and

Similarily, there are subsheaves

of

7i,Bthat rep-

3 ) [ ~ ]with base change resent Q-Gorenstein deformations (wila and 0 ~ / ~ ( !commute for all i E Z). By the Assumption it is well defined on the index one cover of X. Then there is a long exact sequence for Q-Gorenstein deformation of pairs,

o

-+

T$(-B)

-+

T;,~

-+

T;

-+

T ~ ( - B ) ,,;.I. -+

-+

T:, .

180

YONGNAM LEE

Furthermore, if (X,B ) is a stable log surface ( K x + B is ample) then it holds the -1 long exact sequence 0 -+ Tx(-B) 4 + Th by the stability condition. Also -1

and Tx(-B) can be computed by the local-global spectral sequence ([Go], 57.3) : 0 -+ H'(7:) -+ T i ker[Ho(yi) H 2 ( 7 i ) ]-+ 0 0 -+ H 1 ( 7 i ( - B ) ) Ti(-B) + ker[Ho(7i(-B)) -+ H 2 ( 7 i ( - B ) ) ]-+ 0. Let B' be the inverse image of B on the index one cover. Also by Assumption, 0 + Out(-B') -+ O ~--fI OBI -+ 0 gives a resolution of OBI by invertible sheaves. It produces the long exact sequence -+

---f

-+

O -+ T$,B -+ T$

+ H o ( B ,O B ( B ) )-+

T ~ , BT i -+

If H 1 ( B , O ~ ( B= ) )0 then the forgetful functor following commutative digram:

Ti,B

-+

H1(BlOB(B)). is smooth from the

-+

Ha(B,O B ( B ) ) Ti,B Ti lob Lob Lob -2 H 1 ( B , O ~ ( B-+ ) ) T;.B -+ Tx. Since H'(7:) represents deformations of X for which the singularities remain locally a product, our concern is to understand the kernel of the map H o ( F i ) H 2 ( 7 i ) .Assume that X is a locally smoothable semi log terminal surface. Locally smoothability means that for each point p of X there is a threefold X over a discrete valuation ring T with the central fiber ( X , p ) such that W X / T is Q-Cartier and X, is smooth for t E T*. Then H o ( F i )can be computed explicitly. --f

Theorem 2.1. Assume that X is a locally smoothable semi log terminal surface. Then H 0 (y1 X ) = d~Cp+CHo(OD,(Xq))

c

peSing(X,D)

Di

4

where Sing(X, D ) is the set of singular points outside double curves D , a singular point p E Sing(X,D) is the type of A ( l , d p u r- 1) with ( a , r )= 1 [K-SB], Di is a dPT connected component of double curves D , and a singular point q in Di is not locally Q-factorial point. Proof. First recall that the sheaf Ho(?i) = CpeSing(X,D) Def(X,p)

is supported only on singular points and 4')) where q' is a singular point in Di. The type of singular point p in Sing(X, D ) is &(l,dpar - 1) with ( a , r ) = 1, and the dimension of Def(X,p) is d p ([K-SB], Lemma 3.21). Let q' be

+ CDiH0(Oo,(&


a singular point in Di and let X be a Q-Gorenstein smoothing of the germ (X, 4’). If q’ is a Q-factorial point in X then the index one cover of ( X , q ’ ) is smooth with d-semistable central fiber as the notion of F’riedman ( F ] , Definition 1.13). Hence the Q-factorial point q’ does not affect H o ( 7 i ) . If q’ is not a Q-factorial point in X then the preimage q“ of q’ in index one cover is a Q-factorial point. Consider a semistable resolution of index one cover by the succession of weighted blowing of Q-factorial point q”. Then we obtain the proof by the argument in ([F],Corollary 1.12). 0 In some cases, it is possible to compute H 2 ( 7 i ) . Lemma 2.2 is in [Ma].

Lemma 2.2. Let X be a normal surface with log terminal singulairties. Assume that ho(-K,y) # 0 and q(X) = 0. Then H 2 ( 7 i ) = 0.

Proof. Let S be the minimal resolution of X . Then H 2 ( 7 i ) = Ho(s2;2@ K x )L, H o p ? ) = HO(s2,) = 0. 0 Lemma 2.3. Let X be a semi log canonical surface and let Xu -+ X be the normalization. Let (Y,C ) be a pair of component of X u and double curves in Y . If H2(7:(-C)) = 0 for all (Y,C ) then H 2 ( 7 i ) = 0 .

-

Proof. The proof is obtained directly from the short exact sequence, 0 -+ C 7 . ( - C ) + 7; Y

-+

s

0

where S is supported only on double curves. 0

Lemma 2.4. Let X be a smooth surface and E a divisor of X . Assume that

-(Kx + E ) is ample. Then H 2 ( T x ( - E ) )= 0.

Proof. The proof is obtained directly from Akizuki-Nakano vanishing theorem [AN] i.e. H0(s2x(Kx+ E ) ) = 0. 0 Theorem 2.5. Let X be a locally smoothable semi log canonical surface and let p E Sing(X, D ) . Assume that p is permissible. Let ( S ,E ) be the pair of the minimal resolution of the germ ( X , p ) and the exceptional divisor E in S. If H 2 ( 7 s ( - E ) ) = 0

then Ho(7i,,) = C ,

5H2(7$).

Proof. Let p E Sing(X,D ) . Let X be a Q-Gorenstein smoothing of the germ ( X , p ) over a small disk A c C . Since p is permissible, by the succession of weighted X +A blowing up of the singular point p , X admits a semi-stable model y without base change and a general fiber ([Ka], Theorm 1.3). Let Y be the central fiber of J’. It is obvious that the existence of a Q-Gorenstein smoothing of ( X , p ) -+

182

YONGNAM LEE

is the same as the existence of a smoothing of Y . Then Y = ~ ~ = = , where Wk WOis a resolution of the germ ( X , p ) and wk is a rational surface with -(Kwk Dk) ample for k = 1,.. . ,n. Since (Wo,DO)is obtained by blowing up of (S,E ) , f f 2 ( 7 w , ( - D 0 ) )= 0. The proof is obtained by Lemma 2.3 and 2.4. 0

+

3. Compactfications of the space of double cover of P2 branched over curves of degree 4 and 6.

Hacking [Hac] considered the family of compactifications given by moduli space

Pz of log surfaces ( X ,B ) such that K x + crB is semi log canonical singularties and ample where < a 5 1. The compactification is simpler for lower a. He gave a compactification Pd of plane curves of degree d by allowable family of stable pairs of degree d .

Definition. A log surface ( X , B ) is called a stable pair of degree d if X is a proper connected surface, B effective Weil divisor with the following properties : 1. There is a E > 0 such that K X E ) Bis semi log canonical and ample. 2. d K x 3B 0 ( $ K x B 0 if 3)d). 3. There is a Q-Gorenstein smoothing (deformation X of X over a discrete valuation ring T with smooth general fiber such that KX/T, BIT are Q-Cartier).

+

N

+

+ (2 +

N

Define pd(s) = { ( X ,B ) / s 1 allowable family of stable pairs of degree d}. The family ( X ,B ) / S is an allowable family if w t l S , C?X(B)[~I commute with base change for all i.

Theorem 3.1. [Hac] 1. Pd i s a separated proper Deligne-Mumford stack. 2. Pd is smooth if 3 i d . Also the following results a e obtained in [Hac]without difficulty by the definition of stable pair of degree d . Corollary 3.2. [Hac] Let (X, B ) be a stable pair of degree d . Then 1.

- K x is ample,

2. I n d e $ K x 5 d , 3. Let T : U' + U be t h e i n d e x o n e cover. T h e n T'B is Cartier, 4. O B ( B ) )= 0. In particular, if ( X ,B ) is a stable pair of degree d then ( X ,B ) satisfies Assumption in Section 2. In [KL], we proved that if D is a plane curve of degree d and (P2,D ) is a stable pair of degree d then D is a stable plane curve. The converse is also true under a suitable condition. These results make us possible to compare


between the compactifications pd and the compactifications of plane curves by Geometric Invariant Theory. In particular, we obtain the same classification of P 4 as in [Hac] by construction of the stable pairs of degree 4 corresponding the semistable plane curves of degree 4.

Remark 3.3. Consider a plane curve (f = 0) of degree 4 and let ( X ,B ) be a stable pair of degree 4. By the result [KL], X = P2if and only if B is a stable plane curve of degree 4. Hence we need only to consider the strictly semistable cases. The strictly semistable case is a double smooth conic or f ' s with a tacnode ([Mu], 1.12). The stable curve corresponding a double smooth conic is a hyperelliptic curve and its corresponding stable log surface is P(1,1,4) ([Hasl], 4.1). Also the stable log surace corresponding f's with a tacnode is P(l,1 , 2 ) U D P(1,1,2)where D = P1 ([Has2], Section 5) and the singular point of P(1,1,2) is in D. Lemma 3.4. Let ( X ,B) be a stable pair of degree couer Y of X branched over B .

4.

T h e n there is a double

Proof. By Remark 3.3, we need to consider three case. Assume that X = P2 and B is a stable plane curve of degree 4. Then B = 2L where L is a Cartier divisor. Assume that X = P(l,1,4). In this case X is obtained by contraction of -4-section C in the rational ruled surface F4 = P(Op1 @ Opl (4)) and the proper transform B' of B has no intersection with C. Let T : F 4 X = P(1,1,4) be the contraction of C. Then K F ~ -2C - 6 F = r * K x - $' where F is a fiber of F4 -+ P', and 7r*B = B'. Since ( X ,B ) is a stable pair of degree 4,

-

--f

0

-

+ + +

~ * ( 4 K x 3B) 3B' 3B'

+

= 4Kp, 2C = -6C - 24F

Hence, we have B' 2 C f 8 F = 2(C+4F), so B' = 2 L where L is a Cartier divisor. Assume that X = P(l,l,2) u~ P ( l , 1 , 2 ) where D = P'. Then each component 2 = P(1,1,2)of X is obtained by the contraction of -2-section C in F2, and the proper transform B' of B has no intersection with C. Let T : F2 + 2 = P(1,1,2) be the contraction of C. Then K F ~ -2C - 4 F = T*KZ where F is a fiber of F2 Pl, and T*B= B'. Since (X,B) is a stable pair of degree 4, N

N

--f

0

Hence, we have B'

-

-

+ + + +

~'(4K.z 4 0 + 3B)

= ~ K F , 4 F 2 C 3B' = -6C - 12F i3B'.

2 C + 4 F = 2(C+2F), so B' = 2L where L is a Cartier divisor.

184

YONGNAM LEE

Since B’ = 2L for each case, there is a double cover Y’ of P(1,1,4) or P ( 1 , 1 , 2 ) u 0 P(1,1,2) branched over B’. Then the double cover Y of X is obtained by contraction of curves correspoding negative sections.

Lemma 3.5. Let (X, B ) be a stable pair of degree 6. T h e n there is a double cover Y of X branched ouer B.

+

Proof. Since ( X ,B ) is a stable pair of degree 6, K x f BN 0. So QB -Kx, and it is a Cartier divisor on the index one cover. The double cover Y of X branched over B is obtained by consideration of the index one cover. 0 N

iB)

If (X, B ) be a stable pair of degree 4 or 6, then ( X , has semi log canonical singularities. By Lemmas 3.4 and 3.5, there is a double cover Y of X branched over B. Then Y has also semi log canonical singularities ([KO], Proposition 3.16), because K y = f * ( K x i B ) . Since the double curve cannot be a component of B, it is reduced to the case of log canonical singularities.

+

Theorem 3.6. There are compactzfications of the space of double couer of P2 branched over curves of degree 4 or 6. In a future work, we intend to classify the cases in pf3 and intend to compare to this compactification R6 with Shah’s compactification [S] by using Geometric Invariant Theory of plane curves of degree 6.

Acknowledgements The author acknowledges the invitation and the hospitality of Research Institute for Mathematical Sciences, Kyoto University in the summer of 2001, where the work was started. He would like to thank Professors S. Mori and N. Nakayama for helpful discussions during his stay at RIMS. Also he would like to thank Organizing Committee for the invitation of the Conference of Algebraic Geometry in East Asia. REFERNCES [A]

V. ALEXEEV,Boundness and K Z for log surfaces, Internat. 3. Math. 5 (1994), 779-810.

Note on Kodaira-Spencer’s proof of Lefshetz’s theorem, Proc. [AN] Y . AKIZUKIAND S. NAKANO, Jap. Acad. Ser. A 30 (1954), 266-272. [C]

A. CORTI,Semistable bfold flips, Preprint.

[F]

R . FRIEDMAN, Global smoothings of varieties with normal crossings, Ann. Math. 118 (1983), 75-114.

[Gi]

D. GIESEKER, Global moduli for surfaces of general type, Invent. Math. 43 (1977), 233-282.

Topologie Alge‘brique et The‘orie des Faisceau, Hermann, Paris, 1958. [Go] R. GODEMENT,

DEFORMATION THEORY OF SMOOTHABLE SEMI LOG CANONICAL SURFACES

185

[Hac] P . HACKING,A compactification of the space of plane curves, Ph.D. Thesis, Cambridge University, 2001. [Hasl] B. HASSETT,Stable log surfaces and limits of quartic plane curves, Manuscripta Math. 100 (1999), 469-497. [Has21 B. HASSETT,Local stable reduction of plane curve singularities, J. Reine Angew. Math. 520 (2000), 169-194. [Ka]

Y.KAWAMATA, Moderate degenerations of algebraic surfaces, in Complex algebraic varieties Bayreuth 1990 1507 of Lecture Notes in Math. Springer-Verlag, 1992.

[KL] H. KIM AND Y . LEE, Log canonical thresholds of semistable plane curves, Preprint. [KO] 3. KOLLAR,Singularity of pairs, Proc. Symp. Pure Math. 62.1 (1997), 221-287. [K et] J. KOLLARET AL, Flips and abundance for algebraic threefolds, Astbrisque 211 1992. [KM] J. KOLLARAND S. MORI,Birational geometry of algebraic varieties, 134 Cambridge Tracts in Mathematics, 1998. [K-SB] J. KOLLARAND SHEPHERD-BARRON, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988) 299-338. [L]

Y . LEE, Bounds and Q-Gorenstein smoothings of smoothable stable log surfaces, To appear in Contemp. Math.

[Ma] M. MANETTI,Normal degenerations of the complex projective plane, 3. Reine Angew. Math. 419 (1991), 89-118. [Mu] D. MUMFORD,Stability of projective varieties, L’Ens. Math. 23 (1977), 39-110. [R]

Z. Ran, Deformation of maps, in Algebraic curves and projective geometry (Trento 1988), 1389 of Lecture Notes in Math. Springer-Verlag, 1989.

[S]

J. SHAH,A complete moduli space for K3 surfaces of degree 2, Ann. Math. 112 (1980), 485-510.

Department of Mathematics Sogang University Sinsu-dong, Mapo-gu Seoul 121-742, Korea

MODULARCURVES

AND S O M E

RELATED ISSUES

by Viet NguyenKhac Hanoi Institute of Mathematics 1991 Mathematics Subject Classification. Primary llG30, llG05, llG09, 14H10.

Abstract This is an extended exposition of a talk given by the author at the Symposium “Algebraic Geometry in East Asia” held in Kyoto on August last year. The purpose was to collect a unified account of recent results and problems on classical and Drinfel’d modular curves with regards to the d-gonality problem and various relations concerning (directly, or indirectly) strong uniform boundedness conjecture, torsion points, modular parametrization, Seshadri’s constant, model-theoretic aspect, cryptology etc. (a large part of results presented here remained some time as folklore). We discuss also some open problems, related conjectures and perspective.

1. Introduction. The departure point of our investigation was a function field analogue of the so-called strong Uniform Boundedness Conjecture (abbreviated as sUBC). Let @(d)be the set of all isomorphism classes of finite abelian groups occurring as the full groups of torsion in the Mordell-Weil groups of elliptic curves over number fields K with absolute degree [K : Q] 5 d. Then sUBC states that @ ( d ) is finite. After long time efforts of many mathematicians (Yu. Manin, B. Mazur, S. Kamienny, M. Kenku, F. Momose, etc.) the problem was finally solved recently by L. Merel, and also by P. Parent with an effective estimate (cf. [7] and references therein). About five years ago in collaboration with M.-H. Saito we succeeded in carrying out an algebraic proof of a function field analogue of sUBC that was reduced to study the problem of d-gonality of X o ( N ) (see below). As a result it appeared in Preprint Series of Kyoto University ([14]) and also in Duke e-print (alg-geom/9603024). Post factum D. Abramovich communicated to us a n analytical approach (characteristic ‘Institute of Mathematics, P.O.Box 631 Bo Ho, 10000 Hanoi, Vietnam e-mail : socnhimQhn.vnn .vn , nkviet 63t hevinh .ncs t .ac .vn .

187

188

VIET NGUYENKHAC

zero case) essentially due to S.-T. Yau (cf. [24]) strikingly yielding a much sharper bound. Later we became aware of a paper by P. Zograf ([25]), in which the author firstly extends Yang-Yau result ([23]), giving an upper bound for the first nonzero eigenvalue of a compact Riemann surface admitting a meromorphic function of degree d, to the non-compact case. Furthermore P. Zograf proves a precise form of linear bound for the d-gonality problem of modular curves, based on Selberg’s eigenvalue theorem. Henceforth we will quote subsequently this case of the theorem as the Yau-Zograf theorem. Our results were reported on Algebraic Geometry Symposiums at Sendai, Kinosaki, at a meeting of MSJ, circulated among specialists and cited by various authors. Correspondences during this period allowed us to enlarge the paper, in particular this includes an asymptotically good bound (later carried out to the Drinfel’d modular case, see below), a relation with a conjecture proposed in [15], and a complete list of (non-p)torsions of elliptic surfaces over a hyperelliptic base. It should be noted that the analytical proof of Yau-Zograf is true for congruence subgroup case. To proceed algebraically one has to involve the counting point “trick” often used by coding theorists. This trick also allows us to have a complete list of (nonp)torsions of elliptic surfaces over trigonal bases. There is another approach using the class number first proposed in [12], and later completed by F. Momose (Ill]) which turned out to be enough efficient in practice. There might be also a Seshadri constant approach to the d-gonality problem of modular curves which seems to the author quite closely related to the Yau-Zograf theorem. Most recently the d-gonality problem of X o ( N ) found an interesting application in solving the Coleman-KaskelRibet conjecture for modular curves, independently by M. Baker, K. Ribet and A. Tamagawa. On the other hand at the same time one of the authors of [14] realized that our approach is applicable to the case of Drinfel’d modular curves. It was reported on “International Algebraic Conference Dedicated to the Memory of A. G. Kurosh” ([13]). In this instance it poses several questions and related problems. Generally speaking it seems to the author of this paper that it is worth having a unified picture around the ideas exposed here. 2. Function field analogue of sUBC. Let d be a positive integer, and let C be a smooth projective curve defined over a field k of characteristic p # 2,3. Set K = k(C) the function field of C. We say that C is d-gonal, if there exists a finite morphism f : C -+ Pk of degree d. For example, if a smooth curve C is 1-gonal, C is isomorphic to Pi, and if C is 2-gonal, then either g ( C ) 5 1, or C is a hyperelliptic curve of genus g ( C ) 2 2. If C is d-gonal, then there exists an extension of fields

MODULAR CURVES AND SOME RELATED ISSUES

189

-

k ( P k ) = x(t) c-f x ( C ) of degree d. Note that such a field extension may not be unique even if we assume that d is minimal.

Now let E be a non-constant elliptic curve defined over K (non-constancy means that its Klk-trace is trivial). We denote by

E(K)&:

=

{x E E(K)t,,,

:

p not dividing the order of x}

the non-ppart of the torsion subgroup of the Mordell-Weil group. Let @fun(d) be the set of all isomorphism classes of finite abelian groups occurring as the full nonp-parts of torsions in the Mordell-Weil groups of non-constant elliptic curves over function field K with an extension x(P1)c-f x ( C ) of degree 5 d. Then the function field analogue of sUBC states that @fun(d) is finite. Assume the Mordell-Weil group E ( K ) has a torsion element of order N . Then from the moduli theory of Deligne-Rapoport it follows that there exists a finite morphism C -+X o ( N ) onto the modular curve X o ( N ) ,corresponding to the Hecke congruence subgroup, i.e. subgroup in the full modular group PSLz(Z), consisting of matrices

(g 1) with c, divisible by N . Thus the problem is reduced to the

problem of d-gonality of X o ( N ) in view of the following lemma.

Descending Lemma (see, e.g. [14], Lemma 1.3 for a proof valid in any characteristic). Let C1 + Cz be a finite morphism between two smooth projective curves over an algebraically closed field. If C1 is d-gonal, then Cz also is d-gonal. At present there are at least three approaches to d-gonality problem for modular curves: our algebraic approach generalizing Ogg’s method from [17]; class number approach ([12], [ll]);and the analytical approach in characteristic zero (Theorem of Yau-Zograf, [24], [25]). 3. Generalization of Ogg’s method. It is well known (Igusa) that X o ( N ) can be defined over Q canonically and this model has good reduction outside p I N . So by taking reduction at p f N , one can also consider X o ( N ) as the curve defined over lFp. Let v ( N ) denote the number of distinct prime divisors of N . It is known that

[ P S L ~ ( Z: r ) o ( ~=) $] ( N ):= N

n(i+

I/~).

PIN

We have the following theorem (cf. [14]) which gives a purely algebraic proof of function field analogue of sUBC proposed above.

190

VIET NGUYENKHAC

Theorem 3.1. Let X o ( N ) be the Hecke modular curve of level N , and let ko = Q,or IFp with p N . Assume X 0 ( N ) k o is d-gonal, i.e. i f it admits a finite morphism f : X o ( N ) k o IF’%, of degree d . Then one has:

+

-

A) With ko = Q: 1) If N is odd, then (3.1.1) 2) If 3 + N , then

-’”) 1 6

+

2V(N)

3) I n general, as d

B) If ko

= IFp,

&4(N) 12

--$

‘{

00

10d if d = 1’2 6(d - 1)2 + 10 if d 2 3

(3.1.2)

we have the following asymptotic bound

then

+ 2V(N)5 max{p2 + 1 + 2 p . (d

-

1)2, ( p 2

+ 1) . d }

(3.1.4)

For the proof one proceeds as follows. On one hand by counting supersingular points on X O ( N ) F(IFPz) ~ ~ ([17]) for p { N we have the folllowing lower bound for the number IFpz-rational points on X O ( N ) F(,P~P z )

On the other hand for an upper bound we need the “Tower Theorem” (Theorem 3.2 below) with a subsequent application of Weil’s bound. Theorem 3.2 ([14],Theorem 2.1). Let C be a smooth projective curve defined over a perfect field k , f : C + P1z a dominant morphism defined over of degree d . Then there exist a smooth projective curve C‘ and a dominant morphism

f’: c + c’ of degree d’ dividing d , both are defined over k , such that

g(C’) 5 (d/d’ - 1)2

(3.2.1)


191

L e m m a 3.3. There exists a tower of (smooth projective) curves over

satisfying the following conditions:

( i ) hi : Ci +Ci-1 and f, : C --+ Cn are morphisms defined over %, of degrees ei 2 2 and d, respectively. (ii) Putting fi: C 3Ci and d i : = deg f i , we have d = do > d l > ... > dn 2 1 and for each 0 5 i 5 n g ( G ) 5 (d/di - 1)'.

(iii) For every element c E Gk = G a l ( z / k ) morphism fnx has degree d , = deg

fn

f,":c+cnxc;

onto its image.

Theorem 3.2 follows from Lemma 3.3 and a general theory of twisting (or Galois descent). In its turn to get Tower (3.3.1) from Lemma 3.3 (and also bound (3.2.1)) one uses the fundamental inequality of Castelnuovo-Severi. For completeness we formulate this inequality in the following form.

3.4. Inequality of Castelnuovo-Severi ([l],[9], V, and references therein). Assume we are given two surjective morphisms 7ri: C + Ci, i = 1 , 2 between smooth projective curves C, C1, Cz of genera g,g1,92 respectively. Assume further that 7r1 and n2 are independent in the sense that they do not factor via a morphism h: C C' (i.e. there doesn't happen that 7ri = TI 0 h, for certain morphisms ni:C' + Ci, i = 1,2) with deg h > 1. Then

-

9 5 d l g l + dzgz

+ (dl

-

I)(&

-

1)

(3.4.1)

where d i : = deg ni,i = 1 , 2 . Moreover in the case of equality in (3.4.1) all divisors {TI*0 T;(&)}QIC~ o n C1 (respectively {7r2* 0 ~ f ( P ) } p ~ocn ,Cz) are linearly equivalent (but not necessarily form a linear system). For the other formulations of this inequality on languages of correspondences, and also endomorphisms between the corresponding Jacobians, see loc. cit. and [15]. In this connection note a promising conjecture proposed in [15].

192

VIET NGUYENKHAC

4. Class number approach. Let

X := X o ( N ) and let w : = W N be the main

;( il)which normalizes r o ( N ) . It is a

Atkin-Lehner involution defined by

rational involution on X ,so the quotient curve X + : = X / ( w ) is also defined over Q. The number of fixed points of w is given by Fricke's formula

s ( N ) = dnh(-4N),

s:=

where h(-4N) denotes the class number (of quadratic forms of discriminants -4 N ), and 2, if n = 7 (mod 8)

6n =

i: -,

if n~ 3 (mod 8)

1,

otherwise

The idea is to apply the Castelnuovo-Severi inequality to the diagram with 7rz = f the d-gonal map X -+ IF''. We shall analyze below our particular case of cyclic coverings from the linear system point of view, and as the reader can remark, linear system-theoretic argument is also an efficient way in treating Castelnuovo-Severi. Assume X possesses a (base-point free) gi which is invariant under the action of w. It can be seen easily that there is a divisor Do E gfi such that WDO= DO. Putting 7r1

= w,

L:=L(Do):= {f E M ( X ) : (f)+Do 2 0 } , Lo:= {f E L : w*f = f}, L1:= {f E .L : w*f

= -f},

and letting di: = dim&, i = 0,1, one has do + dl = 2. So there are two cases we have to consider: 1) do = dl = 1 and 2) do = 2, d l = 0. From the covering w : X X + it follows that there exists a function y E M ( X ) \ w * M ( X + ) such that 1) y generates M ( X ) over w * M ( X + ) ; 2) w*y = -y and S --f

j=1

where Pj are fixed points of w on X , and E,f is a divisor on X + . Furthermore one can write j=l

with aj = 0, or 1,

C

aj

= d( mod 2) and a divisor Dof

on X f .

193


For f E L1 we have w*(f/y) = f l y , so f /y E w * M ( X + ) .A computation shows S

Df:=( f ) + D o = C aljPj+~-~(Df), j=1

+

where Df' is an effectivedivisor on X + , and a1j is the least residue of aj 1(mod 2). One concludes that the linear system corresponding to Li consists of all divisors of the form C,"=,aijPj w-'(D+), linearly equivalent to DO with D+ effective on X + (here we make a convention aoj : = a j ) .

+

On the other hand, for i = 0 , l we have obviously S

Do =

C

aijPj

+ w-l(D?)

-

(yi),

j=1

where 0 : = E: Thus

+ D l - D' and D' is a divisor on X + supported at {pj':= w(Pj)}. S

degD+ = (d -

C aij)/2. j=1

From the above we see that dim ID+l = 0 if and only if di = 0. Moreover, if dim ID?l 2 1, then one can fix an effective divisor from this linear system, and aijPj w-'(DT)- DO E Li. The map f + f + gives a precise an f :=

c,"=,

+

isomorphism L @ )

N

-

Li. Thus di = dim ID+I.

7w*

The case do = dl = 1 implies both divisors D+, i = 0 , l are effective, and hence one gets s 5 2d (4.1) This also can be seen by using the fibre product argument as follows. If f :X IF' is a d-gonal map w*f = -f , then w induces a 2:1-covering 7r: 'FI P1 so that we obtain a commutative diagram --f

---f

P'Xp1

IF'1

X+

w x+ ?r P1

and X is a normalization of P1 x p X + , where f ' is the induced d-gonal function on X + from Descending Lemma. Hence by adjunction formula we have

+

g ( X ) 5 p , ( P xpi X + ) = g ( X + ) d - 1.

194

VIET NGUYENKHAC

Immediately it is nothing but (4.1) by Riemann-Hurwitz and Fricke’s formulas. This is a particular case of Castelnuovo-Severi. In fact we conjecture that this case never happens for modular curves X o ( N ) and d odd. If true it would give a fair solution of the trigonal classification problem. It should be noted that this problem was settled recently by M. Baker, Y. Hasegawa and M. Shimura with the aid of computer and algebraic method exposed in $3. 4.2. Example. The most “difficult” case in the Ogg’s list [17] is N = 37. In this case g(Xo(37))= 2 , g ( X Z ( 3 7 ) )= l , s ( 3 7 ) = 2 ( d = 2 ) . We have actually degDof = 1, d e g D r = 0, and so do = d l = 1.

Therefore the major case in this approach is the second case (do = 2 , d l = 0). In characteristic zero it can be done as follows (cf. [ l l ] ) .Take a prime divisor q 1 N . Assume that on X o ( N / q ) we also have the second case. Take a non-constant f E Lo(Xo(N/q)). Its pull-back via ‘p: X o ( N ) X o ( N / q ) , for simplicity denoted by the same letter, belongs to L o ( X o ( N ) ) .Thus f is induced by a function on the ex--f

tended upper-half plane which is invariant under r o ( N / q ) ) and so under

(i :)

(N0lq il)) ;(

too. Such a function should be constant: a contradiction.

Hence by the first case on X o ( N / q ) one obtains: s ( N / q ) 5 2d. In positive characteristic class number approach does not give a full answer. At some part one has to combine with the algebraic approach of §3. We restrict ourselves to the following situation. In the notations above assume moreover q { N‘: = N / q (certainly characteristic of the base field char ( k ) { N as assumed before). In any characteristic an elliptic curve E is said to be C M curve, if its endomorphism ring is strictly larger than 25. In terms of pairs let E be an elliptic curve defined over k , and let V N I ,V, be cyclic subgroups of order N‘, q respectively. Assuming the invariance of the action of W N , WN’ on both L spaces of X o ( N ) , Xo(N’) we have the following commutative diagram

( E ,v q + VN’)

1

(E/Vq + VN’?E[q]/Vq+ E“’]/VN’)

b*

lp*

( E )VN’)

(E/Vq+ VNO E“’]/VN‘)

11

(E&, V,

+ Vjv/&)

1W.f

+

(E/Vq E [ N ’ ] E’”’] ,

-k vN//E[N’])


-

-

-

195

v,+

where El: = E/V,+VNi. Actually one can construct a tower ( E ,v)p) (E/V,, V N i /Vq) (E/V,z, Vq2+V” /V,Z) . . . . Thus E is a.C M curve: a contradiction. So again by the first case one gets s(N’) 5 2d. At this point one can also argue altenatively by using reduction modulo q of X o ( N ) (z la Igusa-Deligne-Rapoport ( [ 3 ] )the : reduction of X o ( N ) at q consists of two components, isomorphic to Xo(N’), intersecting transversally at supersingular points, and the involution W N interchanges these components. 4.3. Remark. The effective bound in the class number approach is not so good, since by a well-known theorem of Goldfeld-Gross-Zagier one gets just an exponential bound. Howerver a (non-effective) version of the Brauer-Siege1 theorem gives (asymptotically) “almost” quadratic bound, so in practice this approach is also efficient.

5. Applications. As noted above in characteristic zero S.-T. Yau (unpublished, cf. [24]),and P. Zograf ( [ 2 5 ] )independently found an analytical approach based on the Selberg eigenvalue theorem which treats all congruence subgroups and gives a linear bound for their indexes in PSL2(Z). More generally, let r be a discrete subgroup of PSLz(R) having a fundamental domain of finite area A(r\W) w.r.t. the Poincarh metric on the upper half-plane JHI. The Laplacian A : = - 4 ( I m ~ ) ~ d ~ / d z L E acting on W admits a unique self-adjoint non-negative extension in &(I’\ W): the automorphic Laplacian which we denote by Ar. In the case l? \ W is not compact, the spectrum of Ar consists of a continuous part - the semiaxis [1/4,00), and a discrete part which we order as 0 = A0 < A1 5 - - - .

5.1.Yau-Zograf theorem. In the notation above assume that the Riemann surface I? \ JHI admits a meromorphic function of degree d > 0. T h e n we have either A(I? \ W)< 321rd, or 8Ird A1

0 such that F E S, u E U ==+

e".F E M ,

dM(e".F, F )

< KJuI

ADMISSIBLE VARIATIONS OF MIXED HODGE STRUCTURE

221

Nilpotent Orbit Theorem 191. Let p(s) be the period map of an admissible variation of graded-polarized mixed Hodge structure V -+ A* with unipotent monodromy, and F ( z ) : U + M be a lifting of 4(s) to the upper half-plane which makes (4.1) commute. Then, (1) The limiting Hodge filtration F, of V exists; (2) (F,, N ) is a nilpotent orbit of graded-polarized mixed Hodge structure; (3) There exists constants a , p and K such that Im(z) > a ezN.F, E M

+

and d M ( F ( z ) ,eZN.F,) < KyPe-2*Y,

z = z +iy

Proof. The validity of assertions (1) and (2) was established in Lemma (3.5). To verify assertion (3), assume for the moment that the limiting mixed Hodge structure (F,, ' W ) of V is split over R.Then,

' Y = Y(F,,~w) and Y = Y(F,, W,N) are defined over R (Corollary 3.14); p ( h ) = ' Y - Y is an element of gR (Corollary 3.15); - F, = eiN.F, E M (Schmid's SL2 Orbit Theorem, applied to G r W ) . Moreover, [TY,N ] = -2N since N is a (-1, -1) morphism of (F,, ' W ) . Thus, ~

-

and hence

as ' Y preserves F,. Taking note of the fact that F ( z ) = ezNer(s).F, and exN E G R acts on M by isometries, it then follows from equations (4.7) and (4.8) that

upon setting

e F ( z )= A d ( e i N ) A d ( y i r y ) e ' ( s ) and recalling that ' Y preserves F,. Now, y-;(rY-Y) - f Y ['y,y ] = 0 jy - PY

(4.10)

222

GREGORY PEARLSTEIN

Consequently,

dM(y-&PYeF(z).Fo, y-jpy.F,) = dM(y-3YeF(Z).Fo, y-4y.F,) because ‘Y

-

Y E gw. Thus, by Lemma (4.5): d M (ezNer(s) .F,,

5 y i ( L - l ) d M (eF(”).F,,F,)

e z N F,) .

(4.11)

Invoking Lemma (4.6), one therefore has dM

(erNer(s) .F, , ezN.F, ) 5 K y 4 (L-l)If’(z ) I

(4.12)

To complete the proof in the split over R case, one simply decomposes r(s) according to the eigenvalues of ad ‘Y in order to obtain a bound on the asymptotic behavior of l r ( z ) l . Namely,

lF(z)l = O(y:ce-2TY)

(4.13)

as y -t 00, where c denotes the maximal eigenvalue of ad ‘Y on q-. Inserting (4.13) into (4.12), one obtains the desired estimate dM(eZNer(’).F,, ezN.F,) 5 Kype-2Ty is not split over R. Then, by Theorem (2.11), there

Suppose now that (F,,‘W) exists an element

6 E A-1,-1 (Fco

such that

,Pw

‘w)

(P,, ‘w)= (e-i‘.F,l

q~~,,.~),

is split over R. In particular, upon setting ‘Y = Y = Y(Pm,W,N ) and proceeding as above, one obtains the general proof mutatis mutandis. $5.

The SLz Orbit Theorem

In simplest terms, Schmid’s SL2 orbit theorem asserts that every nilpotent orbit eLN.F of pure, polarized Hodge structure is asymptotic to a 1-parameter map U + D which arises form a representation of SL2 on the associated Lie group Gw. More precisely, one calls a nilpotent Iorbit O(z) = ezN.F an SL2 orbit if and only if there exists a point F, E D and a representation 1c, : S L 2 ( R ) -t Gw such that O(L7.i) = $(g).Fo for every element g E SL2(R).


223

SL2 Orbit Theorem [Ill. Let erN.F be a nilpotent orbit of pure, polarized Hodge structure. Then, there exists (a) an SL2 orbit e Z N . F ;and (ii) a real-analytic function g ( y ) : ( a ,co) + GR, such that y > a ==+ eiYN.F E V ,and (a) eiYN.F = g(y)eiYN.R; (b) g ( y ) and g-'(y) have convergent power series expansions at co of the form Aky-') where A k is an element of gwnker(adN)k+l which maps

(l+xk>O

W ( N ) e into W(N)e+k-l for each index c. In this section, we outline the proof of the following extension of Schmid's SL2 Orbit Theorem:

Theorem 5.2 [lo]. Let e Z N . F be an admissible nilpotent orbit of graded-polarized mixed Hodge structure for which the underlying weight filtration W is of length 2 ( i e . 0 = w k - 2 C wk-1 C wk = v). Then, there exists (i) an admissible sL2 orbit ezN.F; (ii) a real-analytic function g ( y ) : (a,co) -+ Gw; and (iii) an element E gw n ker(ad N ) n A-!>-' such that y > a + eiyN.F E M , and

O Aky-') with Ak E ker(bdN)"l.

Remark 1. In analogy with Theorem (2.20), in order to extend the SL2 orbit theorem to nilpotent orbits of weight length L > 2, one must replace the Gw-valued function g ( y ) by the product of a Gw-valued function gw(y) and an exp(A-'y-l)valued function ex(?'). Remark 2. Although every mixed Hodge structure ( F ,W ) of weight length 2 is split over Iw, and hence equivalent to a direct sum of pure Hodge structures, the corresponding statement for nilpotent orbits is false (i.e. not every such nilpotent orbit is a direct sum of pure orbits). Remark 3. The presentation below is condensed from [lo]. In particular, as consequence of Theorem (5.2) and some auxiliary calculations, one obtains: Corollary 5.3. The holomorphic sectional curvature of an admissible nilpotent orbit eZN.F of weight length 2 is negative, and bounded away from zero whenever y = I m ( z ) is suficiently large. Corollary 5.4. Let eZN.F be an admissible nilpotent orbit of weight length 2. Then, lim e-iyN.~,.vN.F,w) = Y ( F ,W,N ) y-00

224

GREGORY PEARLSTEIN

where Y ( F ,W,N ) is the grading of W constructed by Theorem (3.13).

As in the pure case, the proof of Theorem (5.2) begins with the construction of a connection on a suitable principal bundle P + X . More precisely, let W be a weight filtration of arbitrary length L , and M = M (W,Q , h p l q ) be a classifying space of graded-polarized mixed Hodge structure upon which the Lie group G acts transitively. Define ~

~

~

y ( W )to be the set of all gradings Y of W such that (Y - Y ) ( w k )C Wk-2 for each index k. V to be the direct product of classifying spaces of pure, polarized Hodge structure to which M projects via the map F H FGrW.

X =v x Y(W).

G to be the Lie group consisting of all elements g E G which induce real automorphisms of wklWk-2 for each index k .

Then, there exists a unique projection map rr : X

--f

M with C"-section

Lie(G')-2. (Namely, 7r({HP,q},Y)is the filtration F E M and fiber r - ' ( F ) obtained by lifting the graded filtration FGrW = Halb to the ambient vector space V via the grading Y . )

Lemma 5.6. The space rr : X M is a holomorphic afine bundle upon which the Lie group G acts both transitively and equivariantly. --f

In particular, given a point x E

X ,the associated sequence

defines a principal bundle P ( x ) over

2.

Lemma 5.8. Let F, be an element of M a [cf. 2.101 and xo = a(F0). Then, the vector space decomposition (5.9)

of g = Lie(G) is invariant under the action of Ad(Gxo),and hence defines a connection V on P ( x o ) .

225


Corollary 5.10. Let F, be an element of Mw,x, = a(F,) and D denote the associated connection o n P(x,) defined by Lemma (5.8). Then, given a nilpotent orbit e z N .F such that eiyN.F E M whenever y > a; a n element h, E G which maps x, to a ( e i a N . F ) ; there exists a unique lifting h ( y ) : ( a , c o ) -+ G of a(eiYN.F) t o P(x0) which is tangent to V and satisfies the initial condition h (a ) = h,. Continuing the analogy with the pure case, the next step in the proof of Theorem (5.2) is to derive a system of differential equations which govern the function h ( y ) defined by Corollary (5.10). To this end, given F E M let gc = v+ CB 110 CB 11- CB

denote the vector space decomposition of g@defined by the subalgebras

and observe that F E Mw --r. q+ = +, qo = v0 and A-1,-1 Consequently, the linear operator L : g@+ gc defined by the rule

LIv+ = +i,

Llqo= 0,

=

~-1,-1.

Llq-@*-l,-l = -i

restricts to a linear map L : g -+ g whenever F E Mw.

Theorem 5.11. Let h ( y ) be the G-valued function attached to the base point F, E Mw and the nilpotent orbit e z N . F via Corollary (5.10). Then, h- l dh = -LAd(h-'( Y ) ) N dY

where L is the linear operator defined b y F

(5.12)

= F,.

Now, in order to reduce equation (5.12) to a more tractable form, define

a ( y ) = -2h-'Then,

dh

dY

P(y) = Ad(h-'(y))N

(5.13)

226

GREGORY PEARLSTEIN

Moreover, on account of the horizontality of N = N along eiYN.F and the fact that h-l(y) maps q ; P # N . F , W ) to I&,,w,:

where P+(Y) =

c

P-(Y)

Pol-"Y),

and

=

cP - Y Y )

(5.16)

k>O

k>O

P(y)'+ denote the Hodge components of P(y) relative to (Fo,W ) .

Inserting equations (5.13) and (5.14) into equation (5.12), it then follows that a(y) = a y y )

+ a - y y ) + a + ( y ) + a-(y)

(5.17)

where al>-l(y) = 2

ipyY)

a+(Y) = W+(Y)

a-lJ

= -2q?-lJ(y)

(5.18)

a - ( Y ) = -2iP-(Y)

In particular, upon coupling (5.14) with equations (5.15)-(5.16) and (5.17)-(5.18), one finds that d

&

(st;)

LadZ(y) =

-adX-(y)

(-kX+(2/)

(st:)

(5.19) +

where X+(y) = C u y y ) ,

Z(y) = 2iPOJyy),

X-(y) = a-lJ(y)

form a system of solutions to the monopole equations of d

-2&X+(Y) = [Z(Y),X+(Y)I, d ---z(Y) dY

=

su2,

(5.20)

i.e.

d 2&x-(Y) = [Z(Y),x-(Y)l

[x+(Y),x-(Y)l

(5.21)

Returning now to the proof of Theorem (5.2) proper, let ezN.F be an admissible nilpotent orbit of gradedipolarized mixed Hodge structure for which the underlying weight filtration W is of length 2, and

( f ilr W) = (e-Z6.F,'W)

(5.22)

227

ADMISSIBLE VARIATIONS O F MIXED HODGE STRUCTURE

denote the associated splitting of (F,‘W) defined by Theorem (2.11). Then, the map e ( z ) = e Z N . F is an admissible SL2 orbit of graded-polarized mixed Hodge structure, with base point F, = e i N . P E M~ (5.23) and representation p = y!~+ : s12(R) -+ gw obtained by application of Corollary (3.15) to O(Z). Moreover, because W is of length 2, G = Gw and the associated function P(y) is of the form

Now, in order to collect the associated system of equations (5.18)-(5.21) into a more manageable form, let { e ,f} be a real basis of C2 and

denote the standard matrix representation of su2 relative to the basis

vo

=e

-

v1 = e + if

if,

Then, given a finite dimensional complex vector space V and a finite dimensional representation U of slz(C), contraction against the Casimir element

R

+ 2 x - x + + z2

=2x+x-

of su2 c s/2(C) defines a natural paring

Q : Hom(s1z(C),gl(V))8 H o m ( U , g l ( V ) )-+H o m ( U , g l ( V ) ) via the rule

Q(A, B)(u) = 2[A(X+), B(X-.u)]

+ 2[A(X-), B(X+.u)] + [A(Z),B(Z.u)]

Moreover, upon defining

@(Y)X+ = X+(Y),

WY)Z = Z(Y),

@(Y)x-

= x-(Y)

and

Q(y)v1 = aO1-l (Y),

Q(Y)VO =

equations (5.18)-(5.21) can be written in the compact form

-8@‘(~) = Q(@,@),

-~Q’(Y)=

(5.24)

228

GREGORY PEARLSTEIN

To complete the proof of Theorem (5.2), one must now take up the arduous task of finding explicit series solutions to the non-linear system of equations (5.24) [cf. $6 of [l]].Once this task is complete (approximately 20-30 pages of intricate calculation), one then obtains the desired function g(y), and the constant via the equations h(y) = g(y)y-+p(h)

0, d and x = d ~ ( -19).

X1

+

DEGENERATION OF SL(n)-BUNDLES ON A REDUCIBLE CURVE

231

For any sheaf E on X , P ( E , n ) = x ( E ( n ) ):= dimHo(E(n))- dimH1(E(n))is called Hilbert polynomial of E. If E is a torsion free sheaf on X , and let ri denote the rank of restriction of E to Xi (i = 1 , 2 ) . The rank of E is defined to be

Definition 1.1. A sheaf E on X is called semistable (resp. stable) if, for any subsheaf El c E , one has

The moduli functor Ui : (C - schemes) --+ (sets) was defined as 0s-flat semistable sheaves E on X x S of rank r and x(Elx.) = x for any s E S and it is known that there exists a projective scheme U x , which universally corepresents the functor Ui. For any integer N and polynomial P ( n ) = r(cl+c2)n+x, let W = O X ( - N ) ~ ( ~ ) and W s = 0 ~ ~ s ( - N ) ~Recall ( ~ )that . Quotp(,)(W) is Grothendieck quotient scheme, whose S-valued points may be described as the set of quotients

on X x S , where F is flat over S and its Hilbert polynomial is P ( n ) . Let R""c Quotp(,)(W) (resp. R") be the open set where the sheaf F is semistable (resp. stable) and W s F induces an isomorphism H o (Ws( N ) ) H o( F( N ) ) .Since the set of semistable sheaves with fixed Hilbert polynomial is bounded, we can assume that N is chosen large enough so that: every semistable sheaf with Hilbert polynomial P ( n ) appears as a point in R"". The group S L ( P ( N ) )acts on Quotp(,)(W) and thus on R"". The moduli space UX was constructed as a good quotient --$t

ux = R " " / / S L ( P ( N ) ) . More precisely, the following were known (See [Si] for the more general results)

232

XIAOTAO SUN

Theorem 1.2. ([Se], [Si]) Let U X = R""//SL(P(N)) be the good quotient. Then (1) There exists a natural transformation U i + U X such that U X universally corepresents (2) UX is projective, and its geometric points are

~i.

{

s-equivalent classes of semistable

Ux(@) = sheaves of rank r and degree d

(3) There is an open subset U; c U X , with inverse image equal to R", whose points represent isomorphism classes of stable sheaves. Locally in the e'tale topology on U i , there exists a universal sheaf lUniv such that i f & E @ ( S ) whose fibres &, are stable, then the pull-back of &unav via S + U; is isomorphic to & after tensoring with the pull-back of a line bundle on S . Let Ro C R""be the dense open set of locally free sheaves. For each F E Ro, let F1 = FIX,, F2 = FIX,, we have x(F1) x(F2) = x r and (by semistability)

+

+

Thus Ro is the disjoint union of Rt"xz= { F E 7201x ( F ( x i )= x i } where x1+x2 = x r satisfying (1.1). Let R X 1 ~ X be 2 the Zariski closure of R;"x2in R"" , then

+

has at most r+ 1 irreducible components. Let Uxi be the moduli space of semistable vector bundles Fi of rank r and x(Fi) = x i . It was known that for all possible choices of X I , x 2 satisfying (1.1),R X 1 , X 2 is not empty if U X , and U X , are not empty.

Ux =

u UX1~XZ

=

x1

,xz

u

RXlYXZ//SL(P(N))

x1,xz

has # { ( X I , ~ 2 ) irreducible ) components. Let LObe a line bundle of degree d on X , we define a subfunctor SUL of U i by

S U i ( S )=

Os-Aat semistable sheaves E on X x S of rank r satisfying

W E l x ; ) = p>(Lo)lx; and x ( E l x . { , } ) = x for any s E S

We will prove that there exists a closed subscheme SUx c UX which is reduced, and universally corepresents the moduli functor S U i . To do that, we first prove


233

that the local deformation functor of it is pro-representable. Let A be the category of Artinian local @-algebras. Morphisms in A are local homomorphisms of @-algebras. For any A E A, we denote X x Spec(A) (resp. X o x Spec(A))by X A (resp. X i ) . At any point eo = (W -+ Eo) of the Quot scheme Quotp(,,(W), we have the local deformation functor

G ( A )= {A-flat quotients (WA-H E ) , with x ( E ( n ) )= P ( n ) } such that G(@)= {eo}. It is well known that d ) ~ pro-represents ~ ~ t , ~ ~G. Let LO be a fixed line bundle on X such that det(E0lxo) = COIXO.We can define a subfunctor of G by F ( A ) = { ( W A--H E ) E G ( A )1 d e t ( E l q ) = p > ~ C o } , where ~

X

(resp. O p x , P A ) denotes the projection to X o (resp. X , Spec(A)).

Proposition 1.3. The functor F is pro-representable. Proof. Let A1

-+

A and A2

-+

A be morphisms in A, and consider the map

F(Ai X A A2)

(1.2)

+

F ( A i ) X F ( A ) F(A2).

By Theorem 2.11 of [Sc], it is enough to show that the map (1.2) is bijective when A2 A is a small extension and the tangent space tF is a finite dimensional vector space. But the bijectivity of (1.2) implies that tF has a vector space structure compatible with that of tG (See Remark 2.13 of [Sc]). Thus dzm@(tF)< 00 and we only need to check the bijectivity of (1.2). Let ( W A ~ E l ) E F(A1), (WA --H E ) E F ( A ) ( W A ~ E2) E F(A2) such that the restriction morphisms El 3 E , E2 % E induce isomorphisms -+

-+

-+

(WA1

--H

El)lxA

(WA * E ) ,

(WA2

--H

(WA --H E ) .

E2)lxA

Let B = A1 X A A2, then, since G is pro-representable, there exists a unique (WB-+ &) E G ( B )such that (WB --H I ) I X A ,

(WA1

++

E l ) , (WB * & ) l x a 2

(WA2

* E2)>

where means equality as points of Quot scheme. In particular, there exist morphism & 3 El, 8 3 E2 such that u1. q1 = u 2 .92 and 91, 92 induce isomorphisms E(xAl El, ElxAz E2. By restricting everything to X g and taking wedge product, we have (for i = 1,2)

det(E0)

WqP)

det(E:),

det(E:)

det(u4)

det(Eo)

234

XIAOTAO SUN

satisfying det(uy) . det(q7) = det(u!) ' det(q:). Thus, by Corollary 3.6 of [Sc],

det(Eo) G d e t ( E y ) x d e t ( ~ 0det(E:). ) To prove that det(Eo) 2 p;(. ( C ) , we imitate the arguments of uniqueness in the proof of Proposition 3.2. Since p$.(C)lx;, det(E:), we have morphisms p>. ( C ) 5 d e t ( E y ) , p;(o ( C )

adet(Ei)

which induce the isomorphisms and thus a commutative diagram

det(E:) pz

T

det(u!j')

det(Eo)

(L)

P;r. p1

-

1

det(Ey)

det(u:)

det(Eo)

---+

where 6 is an automorphism of d e t ( E o ) . If there exists a morphism

d e t ( E i ) -% det(E2) such that

det(Eg)

1

det(4)

det(Eo)

82 d e t ( E $ )

-

d 4 4 )

e

1

det(Eo)

is commutative, then 62 has to be an isomorphism (since A2 + A is a small extension) by Lemma 3.3 of [Sc]. Thus we can modify the morphism p;C0(L)3 d e t ( E $ ) to 8-I ~2 : p > o ( ~ 3 ) det(E:) L det(E:),

.5. Thus det(Eo) E p k o ( C ) and we are done if the lift of 6 is always possible. But this is equivalent to the surjectivity of the canonical

so that det(uy) . p l = det(u:)

235


which is true since for any finite dimensional @-algebra A, we have

H o ( X i ,U x ; )

= H o ( X o ,0x0)@cA.

For any point eo = (W ++ Eo),we observe that F has the same singularity with G at eo. To see it, we define a functor T : A --+ Set by T(A) = {Isomorphism classes of A-flat torsion free

@ A)-modules}

such that T((C)= { E o @ ~ x , , , } .There is a morphism of functors 4 : G by €3A). 4((W E A ) )= E A @

--+

T defined

+

It is known that 4 is formally smooth (See Theorem 4.1 of [F2], or Proposition (4.1) of INS]). Here we remark that its restriction to the subfunctor F is also formally smooth.

Lemma 1.4. The morphism 4 : F --+ T is formally smooth. Proof. Let B

--f

A be a small extension] one need to check the surjectivity of F(B)

--t

F(A)

X T ( A )T ( B ) .

For any (WA E A ) E F(A) and N E T(B) satisfying EA @ @ A) S N @ B A, we can find an open cover of X consists two f i n e sets U1, U2 such that zo E U2 \ U1 and U1 @ S p e c ( A ) , (U2 \ (20))@ S p e c ( A ) trivializing the vector bundle E l := EAIX; (See Proposition (4.1) of [NS]). Thus the gluing data of E i is a matrix M E GL(Ux(U1nU2) @A) and E A is obtained by gluing E i and N @ B A . Then the lift EB E F(B) was obtained by lifting the gluing data (See Proposition (4.1) of [NS]). Since S L ( O x ( U 1 n U2) @ B ) S L ( O x ( U 1 n U2) @ A ) is surjective, it is clear that we can choose a lift E GL(Ox(U1n U2) @ B ) of A4 such that the resulting sheaf EB E F(B). This proves the lemma. --ff

--+

Corollary 1.5. The functor F is pro-represented by a reduced semi-normal complete local (C-algebra. Proof. This follows Lemma 1.4 and the study of functor T in [F2] and [Se]. To construct the closed subscheme SUx , which will universally corepresent the moduli functor S U i , we recall some constructions in [Sl] and [S2]. Let 7r : 2 --+ X be the normalization of X and T-'(zo) = {21,52}, then 2 is a disjoint union of X1 and X z (we will identify z1, 2 2 with 20 when we work on X ) . A GPB ( E ,Q) of rank T on 2 is a vector bundle E of rank r on 2 (its restriction to X i is denoted by

236

XIAOTAO SUN

Ei), together with a quotient Exl @ Ex2-+ Q of dimension r . We have constructed the moduli space

P:=

fl

PX1,XZ

XlfXz=X+r of s-equivalence classes of semistable GPB ( E ,Q) on 2 of rank r and x ( E ) = x + r (See [S2]),where Pxl,x2 = {(E, Q ) E P 1 x(Ea) = xi} and X I , x 2 satisfy (1.1). There are also finite morphisms (See [S2])

such that q5 = q5x1,x2 : P 4U x is the normalization of U X . The morphism q5 is defined such that q5( [ ( E Q , ) ] ):= F satisfies the exact sequence

A straightforward generalization of Lemma 5.7 in [Sl] shows that there exists a morphism xl-r(1-91) JF2-r(1-g2) Det : P X l , X Z Jx, +

such that Det([(E,Q)]) = (det(El),det(E2)).Let Li = Colx, (i = 1,2) and

Ll(n1) := L 1 @ 0x1(n151), Lz(n2) := L2 €30x2(n252), where ni = xi

-

r(1 - g i ) - deg(Li).We have the closed subschemes

7;';

:= Det-l((Ll(nl),L2(n2)),

where n1, 722 are determined uniquely by X I , 2 2 satisfying (1.1) since LO was fixed. and define Thus we denote the closed subsets q5x1,x2(P,":;~;) c U p 1 x 2by SUx to be the closed subset

with the reduced scheme structure. Then we have

Theorem 1.6. Let SUx be the closed subscheme of Ux defined above. Then (1) The natural transformation in Theorem 1.2 induces a transformation


237

such that S U x universally corepresents SUL. (2) S U x is a projective, seminormal variety of dimension (r2- l ) ( g - 1). The number of irreducible components of S U x is the same with that of U X , and its geometric points are SUX(C)=

s-equivalent classes of semistable sheaves E of rank r and degree d with det(E(xo)= C O ( X O

(3) There is an open subset SU; c S U x whose points represent isomorphism classes of stable sheaves with jked determinant 130 on X o = X\{XO}. Locally in the &ale topology on SU;, there excists a universal sheaf EUnau such that if E E S U i ( S ) whose fibres &, are stable, then the pull-back of Euniu via S -+ SU; as isomorphic to E after tensoring with the pull-back of a line bundle on S .

Proof. Firstly, it is easy to check (2). In fact, the projectivity and the number of components follow from the construction of S U x . The semi-normality of S U x follows from Corollary 1.5. To show that the set S U x ( @ ) consists of the sheaves F satisfying det(F1xo) = 1301x0,we only need a fact that any line bundles 131, C2 on a smooth projective curve Y satisfy L ~ J Y \ {=~ L2Jy\{y) ) if and only if 131 = C2 8 O y ( k y ) for some integer Ic. To prove ( l ) ,let G be the functor represented by R""and F be the subfunctor defined by F ( S ) = {(Ws--H E ) E G ( S )I det(Elxg) = p>,,Lo}. Let 2 c R"'be the inverse image of SUx and ZZ denote the ideal sheaf of 2.It is enough to show that for any (Ws --H E ) E F ( S ) ,the morphism cps : S 4 R""factors through cps : S + Z c 72"".Namely, one has to prove cp;(Zz) = 0, where cp; : OR=*-+ cps*Os. This is a local problem, it is enough to show that for any s E S the morphism cps," : S = Spec(Bs,s) + R""

factors through Spec(6z,lp,(s.). By Proposition 1.3 and Corollary 1.5, there exists where a complete noetherian local @-algebra R and E = la(&), En

= ( W ~ p e c ( ~ / m--H n ) En) E F ( S p e c ( R / m " ) ) ,

pro-represents the local deformation functor of F at s. Thus cps," : S = Spec(B+,)

+

R""

factors through f : Spec(R) -+ R"", where the pullback f*(WRsB + Euniv ) of the universal quotient is E = l&(En), which is in fact an element of F ( S p e c ( R ) )in our

238

XIAOTAO SUN

case (it is not true for general theory). Thus f = ft : Spec(R) + R""was given by an element ( E F ( S p e c ( R ) ) .Now we can use Lemma 1.7 below, which implies that f,*(Zz) = fi,to prove that fc (and thus cps,+) factors through Z since R is reduced. Having shown (1) and (2), the proof of claim (3) is the same with that of [Si].

Lemma 1.7. Let E E F ( S ) and ft : S R"" be the induced morphism by f&) E Z for any point E S (including non-closed points). -+

E , then

E = (Ws ++ F) E F ( S ) . When S is reduced, we have a stratification S = U S a of S , where S, := {s E Sla(.Fs) = a } are locally closed subschemes (a(F8)was defined by Fs@ f l ) )~:= ::,lf~ ~ @ T?Z$;'-"(~")) ), then using Lemma 2.7 of [Sl] (for S,) we get fc(Sa) c 2. thus proves the lemma. For general S , we use the flattening stratifications S = Si of Mumford (See Lecture 8 of [Mu]) such that the sheaves used in the construction of Lemma 2.7 of [Sl] are flat On Si, then Lemma 2.7 goes through, thus the lemma is proved for general S. Proof. Let

$2 DEGENERATION OF MODULI

SPACE OF SEMISTABLE SL(T)-BUNDLES

Let D be a complete discrete valuation ring with maximal idea m D = ( t ) D and X -+ B = Spec(D) a flat family of proper connected curves. Assume that the generic fibre X, is smooth and the closed fibre XO is the curve X discussed in Section 1. Fix a relative ample line bundle Ox(1) on X such that c3x(l)lxo= Ox(1). Let W x = Ox ( - N ) @ P ( N )and Quotp(,) ( W x )+ B be the relative Grothendieck quotient scheme. For any B-scheme S , we will write XS (resp. Wx,) for X X B S (resp. O X ~ , ~ ( - N ) @ ~Let ( ~RS," ) ) .be the open set of semistable sheaves whose quotient map induces isomorphism O i ( N ) H o ( & ( N ) ) It . is well-known that the relative moduli space of semistable torsion free sheaves is the relative good quotient

RF

fl

-

ux := RS, " / / SL ( P (N))

Let C be a line bundle on X such that Llxo = CO and L, = Clx,. Let

be the closed subscheme of sheaves with fixed determinant L,, and


239

be the moduli space of semistable bundles on X , with fixed determinant C,. Let

(2.1)

c Ry, sux := sux,c ux RTJC,) and SUX, in Ry and UX respectively.

R Y ( C ):= R y p , )

be the Zariski closure of SUx is the relative good quotient of RY(L).We will prove that

7:SUx

Then

+B

universally corepresents a moduli functor SUL, which is defined as

SUL(S) =

i

0s-flat semistable sheaves E on X s := X

XB

S of

rank r and degree d satisfying det(Elx2) = p>oC

where X o := X \ ( 5 0 ) and p x o : X j := X o Actually, we will prove that the functor

XB

S

+

1.

X o denotes the projection.

0s-flat quotients (WX,+ Es)

R k ( C ) ( S )=

on X S such that ES E

SU$(S).

is represented by the closed subscheme R y ( C )c Ry.

Lemma 2.1. The functor R k ( C ) is represented b y the closed subscheme R T ( C )c and the restriction of universal quotient on X X B Ry.

Proof. For any (WX,* Es) E R k ( C ) ( S ) there , is a unique morphism

ps:S+RY such that pullback of the universal quotient (WX,,. X is enough to show that ps is factorized through

s LR Y ( C )

1

--t)

-

Egy) is (WxS* Es).

It

Ry

fl

B=

B = B This is true at the generic fibre, we only need to check the case when S is defined over 0 E B. In the proof of Theorem 1.6 (l),we have show that cps is factorized through (note that we are at the case of X X B S = XO x S )

2 = {(W + E ) E RYoI det(E1p) = p>o(Co)}. Thus the lemma follows the equality: fL1(0) := R Y ( C ) o = 2,which we will prove in the next lemma.

240

XIAOTAO SUN

Lemma 2.2. f ~ ' ( 0 ):= R Y ( L ) = ~ Z.

Proof. We first prove that Ry(C)o c 2. Let t o = (W ++ &o)E Ry(L)o,we need to show that det(Eo1xo) = P > ~ ( ~ O ) . By the definition of R","L), there exists a complete discrete valuation ring 5 dominant D,and a morphism T := Spec(5) -% R$?(L)over B such that

Namely, there is a OT-flat torsion free sheaf & on X X B T such that det(&lx,,) = L, and &lxo= &o. Let w : ?,T + XT be the desingulaxization of XT at 2 0 . The exceptional divisor LZ-~(Q) = Ei is a chain of (-2)-curves, and the special fibre 2 0 of 2~-+ T at 0 is XI X2 CEi, and XI n ( X 2 C E i ) = {a}, X2 n (XI CEi) = {m}. Identifying 2~\ P - ~ ( z O ) % XT \ {ZO} = X;, we can extend det(&lx;) into a line bundle det(&lx;) on XT, which satisfies that (a*pkL)Izv= det(EI,p)ITq. Thus one has WEIx;) = ( W * P X ) €3 O&(V),

+

+ +

+

5

where V 20is a vertical divisor. Since O ~ ~ ( 2is0trivial, ) there are integers such that OzT(V)Ixl = O ~ ~ ( n l z OzT(V)Ix2 l), = Ox2(n2z2).Hence

n1,

n2

We are left to prove Z c Ry(L)o. Since the locus Ro c f - l ( O ) = RFo of locally free sheaves is a dense open subset of FLYo,we only need to check that

Ron 2 c R y ( L ) o Let 60 = (W Eo) E Ro n 2, then &O is a vector bundle on XO det(EoIx0) = P > ~ ( L Owhich ) , implies that (for i = 1,2) ++

=zlx0.

=

X and

Letz=C@Ox(XI+(nl+l)X2)and~o Then,sinceOx(Xl+X2) S O X , we have O x ( X 1 (nl + l)X2)lxi = Oxi(nizo) and z0lxi = det(Eo)Ixi, which implies that - det(E0) = Lo = L1x0.

+


24 1

Let di = deg(€oJx,) and J?, (i = 1,2) be the Jacobian variety of line bundles of degree di on X i . Let J(X,) be the Jacobian varirty of line bundles of degree d on X,. Then J(X,) can be compactified into a relative Jacobian variety

J(X)*B

J21

such that its special fibre J ( X ) o = x J%2 (Otherwise, we can modify J ( X ) through an isomorphism by tensoring a suitable line bundle Ox(IclX1 IczX2) on X ) . Let R$ c 722 be the locus of locally free sheaves. By taking determinant, we have a morphism

+

The line bundle I? on X gives a section a : B + J ( X ) such that a(0) = a ( B )n J ( X ) o . Let R" := D et -' (o(B)) + B and €" = € u n i v ( x x B ~Then -.

d e t ( Y ) =p>(I?) @ p G C ( K ) for some line bundle K on R ', and &, E R ' . Thus, for any open set that K is trivial on U B , we have U n Ry(C,) # 8. Thus

60 E W such

--f

(0

RSSX(L,),

which proves the lemma. Let A x be the category of Artinian local D-algebras. For any point

e = (W --H €0) E

R2((L)o,

let F x : A x + Set denote the local deformation functor of R Y ( C ) at e. Similarly, we define a functor TX : A + Set by Tx(A) = {Isomorphism classes of A-flat torsion free (Ox,,, @D A)-modules} i

such that T ( C ) = (€0 @ Ox,zo},There is a morphism of functors

defined by ~ ( ( W X --H ExA)) = ExA @ (Ox,,, @LI A ) . Similar with Lemma 1.4, we have

242

XIAOTAO SUN

Lemma 2.3. The morphism $I : FX

--+

TX is formally smooth.

Proof. The same arguments with Lemma 1.4, we just remark an easy fact: Let B be a flat D-algebra and M be a B-module, flat over D. Assume that M @ D D / m is a free BlmB-module. Then M is a free B-module.

Let X = matrics, and

and Y = (yij)(r-a)x(T-a)be the ( r - a ) x ( r - a ) 2 = Spec

D[X,YI ( X . Y -t,Y. x -t)'

Let 0 E 2 be the point defined by the idea (X, Y ,t ) O Z . Then we have Lemma 2.4. Let e = (W + &o) E R Y ( L ) o such that a(&o) and e2 such that

el

= a.

Then there exist

Proof. This is the consequence of Lemma 2.3 and the results of [F2]and [NS].

Theorem 2.5. Let X + B be a regular scheme with closed fibre X O = X and a fixed relative ample line bundle O x ( 1 ) . Let C be a line bundle on X such that Clx, = LO and let S U x + B be defined as (2.1), Then

7:

( 1 ) There is a natural transformation SU)l universally corepresents SUxb . (2)

7: S U x

--f

--+

S U x such that

7: S U x

--+

B

B is a flat family of projective varieties of dimension (r2 -

l ) ( g - l), whose general fibres f - ' ( b ) are moduli spaces SUxb) of semistable vector bundles of rank r and degree d with fixed determinant Cb = Clx,, and its closed fibre fl'(0) is the variety SUx in Theorem 1.6.

Proof. ( 1 ) follows from Lemma 2.1 and the same arguments of [Si]. ( 2 ) is clear. REFERENCES

[Fl] G. Faltings, A prooffor the Verlinde formula, J. Algebraic Geom. 3 (1994), 347-374. [F2]

G. Faltings, Moduli-stacks for bundles o n semistable curves, Math. Ann. 304 (1996), 489515. [Gi] D. Gieseker, A degeneration of the moduli space of stable bundles, J. Differential Geom. 19 (1984), 173-206. [GLl] D. Gieseker and J. Li, Irreducibility of moduli of rank 2 vector bundles o n algebraic surfaces, J . Differential Geom. 40 (1994), 23-104. [GL2] D. Gieseker and J. Li, Moduli of high rank vector bundles over surfaces, J. Amer. Math. SOC.9 (1996), 107-151.


243

[Kal] Ivan Kausz, A modular compactification of the general linear group, Documenta Math. 5 (200), 553-594. [Ka2] Ivan Kausz, A Gieseker type degeneration of moduli stacks of vector bundles o n cumes, Preprint (2001), 1-59. [Li] K. Z. Li, Lectures o n moduli theory ( i n Chinese, unpublished) (2001). [Mu] D. Mumford, Lectures o n curves o n a n algebraic surface, Annals of Math. Studies 59 (1966). [NR] M.S.Narasimhan and T.R. Ramadas, Factorisation of generalised theta functions I, Invent. Math. 114 (1993), 565423. [NS] D.S. Nagaraj and C.S. Seshadri, Degenerations of the moduli spaces of vector bundles o n cumes I, Proc. Indian Acad. Sci.(Math. Sci.) 107 (1997), 101-137. [NSe] D.S. Nagaraj and C.S. Seshadri, Degenerations of the moduli spaces of vector bundles on cumes II, Proc. Indian Acad. Sci.(Math. Sci.) 109 (1999), 165-201. [Sc] M. Schlessinger, Functors of A r t i n Rings, Trans. of AMS. 130 (1968), 208-222. [Se] C.S. Seshadri, Fibre's vectoriels sur les courbes algkbriques, Asterisque 96 (1982). [Si] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I , I.H.E.S. Publications Mathematiques 79 (1994), 47-129. [Sl] X. Sun, Degeneration of moduli spaces and generalized theta functions, J. Algebraic Geom. 9 (2000), 459-527. [S2] X. Sun, Factorization of generalized theta functions at reducible case, math. AG/0004111 (2000), 1-33. INSTITUTE O F MATHEMATICS, ACADEMIA S I N I C A , BEIJING 100080, CHINA E-mail address: [email protected]

DEPARTMENT OF MATHEMATICS, THEUNIVERSITY OF HONGKONG, P O K F U L A M ROAD, HONG KONG E-mail address: [email protected]

Refined Brill-Noether Locus and

Non-Abelian Zeta Functions for Elliptic Curves Lin WENG Graduate School of Mathematics, Kyushu University, Japan In this paper, new local and global non-abelian zeta functions for elliptic curves are defined using moduli spaces of semi-stable bundles. To understand them, we also introduce and study certain refined Brill-Noether locus in the moduli spaces. Examples of these new zeta functions and a justification of using only semi-stable bundles are given too.

1. Refined Brill-Noether Locus 1.1. Moduli Space of Semi-stable Bundles 1.1.1 Indecomposable Bundles Let E be an elliptic curve defined over &, an algebraic closure of the finite field F, with q-elements. Recall that a vector bundle V on E is called indecomposable if V is not the direct sum of two proper subbundles, and that every vector bundle on E may be written as a direct sum of indecomposable bundles, where the summands and their multiplicities are uniquely determined up to isomorphism. Thus to understand vector bundles, it suffices to study the indecomposable ones. To this end, we have the following result of Atiyah [At]. In the sequel, for simplicity, we always assume that the characteristic of F, is strictly bigger than the rank of V . Theorem. (Atiyah) (a) For any r 2 1, there is a unique indecomposable vector bundle I, of rank r over E , all of whose Jordan-Holder constituents are isomorphic to c 3 ~ Moreover, . the bundle I, has a canonical filtration (0)

c F 1 c . . . c F'

= IF

with Fi = I* and F i f l / F i = 0 ~ ; (b) For any r 2 1 and any integer a , relative prime to r and each line bundle X over E of degree a , there exists up to isomorphism a unique indecomposable bundle W,.(a; A) over E of rank r with X the determinant;

245

246

LIN WENG

(c) The bundle I T ( W T / ( aA)) ; = I , @ W,!(a; X)is indecomposable and every indecomposable bundle is isomorphic to I,(W,!(a; A)) for a suitable choice of r,r‘,A. Every bundle V over E is a direct sum of vector bundles of the form ITi(WT;(ai; Xi)), for suitable choices of ri,r: and Xi. Moreover, the triples (ri,r:, Xi) are uniquely specified up to permutation by the isomorphism type of V. Here note in particular that W,(l,X) zli A, and that indeed I,(W,#(a; A)) is the unique indecomposable bundle of rank rr‘ such that all of whose successive quotients in the Jordan-Holder filtration are isomorphic to W,/ (a;A). 1.1.2. Semi-stable Bundles As above, let V be a vector bundle over E. Define its slope p(V) by p(V) := deg(V)/rank(V). Then, following Mumford ([Mu]), V is called stable (resp. semistable), if for any proper subbundle W of V, p ( W ) < p(V) (resp. p ( W ) _< p(V)). For example, I T ( W T J ( aA)) ; is semi-stable with p ( I T ( W T t ( a ; X ) = ) ) a/r’. Theorem. (Atiyah) (a) Every bundle V over E is isomorphic to a direct sum @iV, of semi-stable bundles, where p(V,) > p(V,+l); (b) Let V be a semi-stable bundle over E with slope p(V) = a/r’ where r’ is a positive integer and a is an integer relatively prime to r’. Then V is a direct sum of bundles of the form I T ( W T l ( a ; X ) )where , X is a line bundle of degree a. 1.1.3. Moduli Space of Semi-stable Bundles Let V be a semi-stable vector bundle, then we may associate it a Jordan-Holder filtration, which is far from being unique. However the associated graded bundle, denoted as gr(V), is unique. Following Seshadri, two semi-stable vector bundles V and V’ are called S-equivalent, denoted by V -S V’, if gr(V) N gr(V’). Now set

ME,,(X) = {V : semi - stable, detV = X,rank(V) = r } / NS . Then from the above classification we have the following well-known Theorem. (Atiyah, Mumford-Seshadri) With respect to a fixed pair (r,A), there exists a natural projective algebraic variety structure on Mg,,(A). Moreover, i f X E Pico(E), then ME,,(X) is simply the projective space PC-l. F’l

1.2. Refined Brill-Noether Locus 1.2.1, Rational Points Now let E be an elliptic curve defined over a finite field F,. Then over B = E XF,K, from 1.1.3, we have the moduli spaces ME,,(X) (resp. ME,,(^)) of semistable bundles of rank r with determinant X (resp. degree d ) over E . As algebraic varieties, we may consider F,-rational points of these moduli spaces. Clearly, by definition, these rational points of moduli spaces correspond exactly to these classes of semi-stable bundles which themselves are defined over F,. (In the case for

247

REFINED BRILL-NOETHER LOCUS AND NON-ABELIAN ZETA FUNCTIONS

M Z , ~ ( X X) , is assumed to be rational over Fq.) Thus for simplicity, we simply write M E , ~ ( Xor) b t E , r ( d ) for the corresponding subsets of F,-rational points. For example, we then simply write Pico(E) for Pico(E)(Fq).And by an abuse of notation, we often call these subsets the moduli spaces of semi-stable bundles too. 1.2.2. Standard Brill-Noether Locus Note that if V is semi-stable with strictly positive degree d, then ho(E,V) = d. Hence the standard Brill-Noether locus is either the whole space or empty. In this way, we are lead to study the case when d = 0. For this, recall that for X E Pico(E), M E , ~ ( X=) {V : semi - stable,rank(V) = r, det(V) = A}/

NS

is identified with = A, ~i E Pico(E),i = I , . . . , r } /

{V = @=1Li :

Niso=

pr-I

where / -is0 means modulo isomorphisms. Now introduce the standard Brill-Noether locus

w&-(A):= {[Vl E M E , ~ ( X: ho(E, ) gr(V)) 2 a } and its 'stratification' by w g , r ( X ) o :=

{[vl

E W E , r ( X ) : ho(E,gr(V))= a } =

wg,r(A)\Ub?a+l wk,r(A)

One checks easily that W&(X) !x P(T-a)-l. Thus in particular, we have the following Lemma. With the same notation as above,

W;>y1(X)

N

W&(A),

and

WEa, rf +' l (A)'

1~

W&.(X)o.

1.2.3. Refined Brill-Noether Locus The Brill-Noether theory is based on the consideration of ho. But in the case for elliptic curves, for arithmetic consideration, such a theory is not fine enough: not only ho plays an important role, the automorphism groups are important as well. Based on this, we introduce, for a fixed ( k 1)-tuple non-negative integers (ao;a l , . . . ,a k ) , the subvariety of W z r by setting WZFl .,a k (A) := {[VI E w ~ ~ :(gr(V) x ) = 0;') B~LQ"' = A, ~i E Pico(E), i = 1 , ..., k } .

+

1..

~

f

=

~

~

i

~

l

)

,

248

LIN WENG

Moreover, we define the associated ‘stratification’ by setting

w;”’ .,ak (A)’ I..

:=

{[v] E w;pr’”“’ak((x), # { O E , L ~ ,. . . , L k } = k + 1).

From definition, we easily have the following Lemma. With the same notation as above, w,,r++l+lial,.

and

..,ak

wzr+;y1,...,a&(X)o

ao;ai ,...,ak wE,r

(4

Wao;al,...,ak E ,T

(4’.

Moreover, ao;ai,...,a&

M E , r ( X ) = uao;ai,...,a k W E , ,

(X)O,

where the union is a disjoint one. In fact, the structures of W ~ ~ ” . . ” a kcan ( X )be given explicitly: They are products of (copies of) projective bundles over E and (copies of) projective spaces. Proposition. With the same notation as above, regroup (ao;a l , . . .,a k ) as (ao;b?’), . . . ,b,(”‘))with the condition that bl > b2 > . . . > bl and s1, s2,. . . ,s1 E Z>O,then (1) if bl = 1, 1-1

(A)

w;p~l’...’~k N

JJp2-1 x PSI; i= 1

(2) if bi > 1,

Proof. This is because we have the following two facts about the quotient of t products of elliptic curves: (1) The quotient space E(n)/Snis isomorphic to the P”-l-bundle over E ; and (2) The quotient of E(”-l)/S, is isomorphic to P(,-l). Here we embed E(,-’) as a subspace of E(”) under the map: (51,. . . ,%) t-+ (21,. . . ,&-1,2n) with zn = X - (21 22 . . . $,-I). We end this subsection with the following intersection theoretical discussion. For simplicity, let X = OE. Then from above, we have the refined Brill-Noether loci w ~ ~ ” ‘ . ” a( kO E )which are isomorphic to products of (copies of) projective bundles over E and (copies of) projective spaces. Thus it would be very interesting to see the intersections of these special subvarieties in M E , ~ ( O E =)PrP1. For this

+ + +


249

purpose, define the so-called Brill-Noether tautological ring B N E , ~ ( O Eto ) be the subring generated by all the associated refined Brill-Noether loci. For examples, (1) If T = 2, then this ring contains two elements: 1-dimensional one W 2.0 E ' , 2 ( O=~ ) {[UE@ O E ] )and the whole P'; (2) If r = 3, then (generators of) this ring contains five elements: 2 of 0-dimensional objects: wz,:( 0 ~=){ and W;,: ( O E )= { [OE@ TJ2)] : T2 E Ez} containing 4 elements; 2 of 1-dimensional objects: WZ,:' = { [C?E@L@L-']: L E Pico(E)} N pl,a degree 2 projective line contained in p2= M E , ~ ( o E ) ; and w OE,3 ; ~ J = { [ L ( 2@ ) LL2]: L E Pico(E)} a degree 3 curve which is isomorphic t o E ; and finally the whole space. Moreover, the intersection of Wz,kl = P1 and W:,?' = E are supported on 0-dimensional locus WZ,?', with the multiplicity 3 on the single point locus W ~ , : ( O Eand ) 1 on the complement of the points in Wi,?'.

[o;']}

2. Measure Refined Brill-Noether Locus Arithmetically

2.1. Invariant a In the rest of this section, we use the same notation as in 1.2. To measure the Brill-Noether locus, we introduce the following arithmetic invariant ~ E , ~ ( Xby ) setting

Also set

Before going further, we remark that above, we write V E [V]in the summation. This is because in each S-equivalence class [V], there are usually more than one vector bundles V. For example, log'] consists of Og), Og) @ 1 2 , 12 @ 1 2 , OE @ 4 , and 1 4 by the result of Atiyah cited in 1.1. Thus, by Lemma 1.2.3, we have the following Lemma. With the same notation as above,

We end this section with the following

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LIN WENG

Conjecture. For all A E Pico(E),

2.2. Invariants /3 and y Due to the importance of automorphism groups, following Harder-Narasimhan, and Desale-Ramanan, we introduce the followingP-series invariants p ~ , ~ ( d ,BE,,.(A) ), and p z F l ? . . . ? a(A) k by setting

and

Corresponding to the Conjecture 2.1.1 for a!, for p, we have the following deep Theorem. ([HN] & [DR]) For all A, A’ E Picd(E),

Moreover,

Here Nl denotes #E(:= #E(F,)) and < E ( s ) denotes the Artin zeta function for elliptic curve EIF,. Thus, we are lead to introducing the y-series invariants ~E,,.(A) and y’$!l;al’..’’ak(4 b y setting y := a! - p. That is to sya,


251

and

Clearly, by the above Theorem, the Conjecture 2.1.2 for a is equivalent to the following Conjecture. or all x E Pico(E),

The advantage of this Conjecture is that now the support of the summation is over Wk,?(X),a codimension 1 projective subspace. Similarly, we have the following Lemma. With the same notation as above, for x E Pico(E),

c and

3. New Non-Abelian Zeta Functions for Elliptic Curves 3.1. Non-Abelian Local Zeta Functions 3.1.1. Definition Let E be an elliptic curve defined over F,, the finite field with q elements. Then we have the associated (F,-rational points of) moduli spaces ME,,(d). By definition, the rank r non-abelian zeta function [E,,-,F,(s) of E is defined by setting

Here d(V) denotes the degree of V. Remark. We call the above infinite sum the rank r non-abelian zeta function because when r = 1 the above summation C E , ~ ( S )coincides with the classical Artin

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LIN WENG

zeta function for the elliptic curve E over F,. In other words, the classical Artin zeta function C E ( S ) may be written as

where d ( L ) denotes the degree of L . 3.1.2. Basic Properties With the above definition, by a direct yet long calculation, we have, by setting t = q-' and Z , , , F q ( t ) := < E , ~ , F , ( s ) ,the following Fundamental Identity. Let E be an elliptic curve defined over F,. Then f o r any I.

E

z>o,

Here

Remark, We in this paper choose not to write down the detailed elementary calculations, despite the fact that some of them are very long and sometimes a bit complicated. As a direct consequence of this Fundamental Identity, we have the following Theorem. Let E be an elliptic curve defined over F,, the finite field with q elements. Then the associated rank r non-abelian zeta function <E,~,F,( s ) satisfies the following basic properties: (1) (Rationality) ZE,,.,F,(~)may be written as the quotient of two polynomials; (2) (Functional Equation) CE,~,F, ( s ) = <E,r,F, ( 1 - s ) . From here we may conclude that after suitable arrangement, the product of two reciprocal roots of P E , ~ , (t), F , a degree 2r polynomial, are always equal to q, the cardinal number of the base field. Moreover, we know that up to the term Y E , ~ ( O ) , from Theorem 2.1.2, the coefficients of these local non-abelian zeta functions can be computed. 3.2. Non-Abelian Global Zeta Functions

REFINED BRILLNOETHER LOCUS AND NON-ABELIAN ZETA FUNCTIONS

253

3.2.1. Definition. Now let E be an elliptic curve defined over a number field F . For a fixed positive integer r, set s b d to be the union of a11 infinite places (of F ) , d l finite places where E have bad reductions, or where the characteristics of the residue fields are less than r. By definition, a (finite) places u of F is good, if u is not in

Shad. Thus, in particular, for good places u , by taking reduction of E at v, we have the associated regular elliptic curve E, defined over F ( u ) N Fqu,the residue field of F at u, where qv denotes cardinal number of F ( u ) . Then by applying the construction of 3.1, we get the rank r local non-abelian zeta functions <E,,,-,F~,(s). In particular, we further obtain, by the rationality, the corresponding polynomials PE,,,.,F~(~) of degree 2r (with 1 as constant terms). Definition. Let E be an elliptic curve defined over a number field F . Then for any positive integer r , define its associated rank r global non-abelian zeta function < E , ~ , F( s ) by setting

v:good

Here qv denotes the cardinal number of the residue field of F at u. Clearly, if r = 1, this then recovers the famous Hasse-Weil zeta function for elliptic curves, for which we have the celebrated BSD conjecture. Thus it seems to be quite natural for us to call the above Euler product a global non-abelian zeta function for elliptic curve. Surely the biggest problem we are now facing in this algebraic part of our nonabelian zeta function is to give the precise region over which the Euler product in the definition converges. For this we have to use our refined Brill-Noether theory discussed in Section 1. 3.2.2. Estimations for ,f3 and y We now want to give estimations for invariants p and y. So let E be an elliptic curve defined over a finite field F, as before. First we study PQ-. For this, following Harder-Narasimhan, we interpret the Tamagawa number of SL(n) is 1 as follows: Proposition. (=[DR Proposition 1.13) Let < ~ ( s be ) the Artin zeta function of E . Then for any jixed X E Picd(E),

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LIN WENG

Therefore, for a fixed X E Picd(E), 1

c

V:rank(V)=r,det(V)=X #Aut(V)

= O(q-1).

This implies in particular that PE,r(X)

= O(q-').

Thus, by Hasse's result ([Ha]) on N1 = #Picd(E), we know Nl = O(q). Therefore we complete the following Proposition I. With the same notation as above, , / 3 ~ , ~ (=d O(1), ) as q

--$

00.

Next we study ~ E , ~ ( O )As . our final purpose is to give a good estimation for the coefficients of P E , ~ , F , hence ( ~ ) , by the Fundamental Identity in 3.1.2, it suffices to give a lower bound for ~ E , ~ ( O )For . this, we consider semi-stable vector bundles V with gr(V) = OE @ @:IiLi with Li E Pico(E) and # { O E , L ~. ,. . , L r - l } = T . Clearly then V = gr(V) and there are totally O(N;-') or better O(q'-l) of them, as q + 00 by the above mentioned result of Hasse. On the other hand, easily, we have ho(V) = 1 and #Aut(V) = (q - l)r.Therefore

This then implies the following Proposition 11. With the same notation as above, y ~ , ~ (=d O(l), ) as q + 00.

3.2.3. Convergence of Global Non-Abelian Zeta Functions As above, let E be an elliptic curve defined over F,. Then by the Fundamental Identity and the Functional Equation for local non-abelian zeta functions of elliptic curves, r

i=l with Ai E R. So, as q + 00, by Proposition I and 11 in 3.2.1, i

C AiAj =O(q2); ...............

n r

Ai =O(qr).

i=l


255

So, Ai = O(q),i = 1 , . . . ,r. As a direct consequence, we then have the following Theorem. Let E be an elliptic curve defined over a number field F . Then all rank r global non-abelian zeta functions <E,~,F(s),defined by (infinite) Euler products converge when Re(s) > 2. Remark. When r = 1, by the above mentioned result of Hasse, the Hasse-Weil While this zeta functions, i.e., <E,~,F(s)in our notation, converge when Re(s) > result is better than ours in the case when r = 1, we are also quite satisfied with our one as our Theorem is the best possible for general r at this stage. (See e.g. 4.1.1 below.) Now, a natural question is whether our rank r global non-abelian zeta functions have meromorphic extensions and satisfy the functional equation. We believe that it should be the case. In fact we have the following

2.

Working Hypothesis. By introducing also factors for bad places, the completed rank r non-abelian zeta functions ~ E , ~ , F ( sfor ) elliptic curves have meromorphic continuation to the whole complex plane and satisfy the functional equation

Clearly, we then would also hope for certain type of such zeta functions, the inverse Mellin transform would lead to modular forms of fractional weight 1 :. Unfortunately, we have not yet obtained any examples to support this speculation. But if it holds, we then have a systematical way to construct fractional weight modular forms, for which, except in the case of half integers we know very little.

+

4. Examples and Justifications

4.1. Lower Rank Non-Abelian Zeta Functions 4.1.1. Rank Two Let E be an elliptic curve defined over the finite field F,. If rank r is two, we need then only to calculate P ~ , 2 ( 0 ) , P ~ , 2 (and 1 ) TE,r(O). We first consider P ~ , 2 ( 0 ) .Then by our discussion on Brill-Noether locus, it ). suffices to calculate P E , ~ ( ~ ENow

Clearly,

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LIN WENG

consisting of just 1 element; W~,;(UE)O= {[V] : gr(V) =Ti2),T2 E E2,T2 # UE} consisting of 3 elements coming from non-trivial while

T2 E E2,

2-torsion subgroup of E ;

is simply the complement of the above 4 points in P'. With this, one checks that

+(q

+1

-

(3

+ 1)) . (4-1 ~

--q + 3 q2

And hence

- 1' P E , 2 ( 0 ) = Nl

q+3 . 42.

As for P ~ , 2 ( 1 )it, is very simple: Any degree one rank two semi-stable bundle is stable. Moreover, by the result of Atiyah cited in 1.1, there is exactly one stable rank two bundle whose determinant is the fixed line bundle. Thus

Finally, we study YE,2(0). We want to check Conjecture 2.1.2. Clearly if is supported on

# OE, then YE,2(X)

W:,;(A)

=

{[v]: gr(v) = OE ex}

consisting only one element with V = gr(V) = OE @ A. So YE,2(A) =

q-1

1

-= q - 1' (q - 1 ) 2

On the other hand, YE,z(OE) is supported on


257

consisting only one element too. However, now in the single class [V] with gr(V) = U g ) ,there are two elements, i.e., U g ) and I,. So

Thus we have checked all conjectures in the case r = 2. Proposition. With the same notation as above, in the case r = 2, all conjectures in this paper are confirmed. I n particular, ZE,2,F,

N1 q-1

( t )= -'

1

+ (q - l ) t + ( 2 q - 4)t2 +

(42 -

q)t3

+ q2t4

(1 - t2)(1 - q2t2)

We reminder the reader that in this case, P E , 2 , F q ( t ) is independent of E and with integer coefficients. Thus for global rank two non-abelian zeta function, we obtain an absolute Euler product, say in the case when the base field is Q,

Ez(s) =

n

1

+

p23,prime 1 (P - 1 k S

+ (2P - 4)P-2" + (p2 - p)p-3s + p2p-4s Re(s) > 2.

Clearly we expect more from such a beautiful Euler product. In fact, recently, in a private communication, Andrew Booker shows that the produt could be meromophically extended to the region Re(s) > 1. 4.1.2. Rank Three First, we check that

By the fact that Aut(UE @ 1 2 )

= ( q - l)'q3,

the above identity is equivalent to

1

1 43 - 1 ++ q2 - 1 + q - 1 (q2 - 1>(q2- 9.1 (4 - 1)q (!I3 - 1>(q3- q ) ( q 3 - q2) ( 4 1I2q3 ( 4 - l)q2' -

which may be directly checked. This together with the similar relation for r = 1 discussed in 4.1.1 leads to

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LIN WENG

so we are left to study P ~ , 3 ( d )d, = 0, 1 , 2 . Easily, we have

since here all semi-stable bundles become stable. Thus we are lead to consider only P ~ , 3 ( 0 ) . So it suffices to give PE,~(X)for any X # U E . (Despite the fact that PE,~(X= ) PE,~(C?E)for any X E Pic'(E), in practice, the calculation of PE,~(X) with X # C?E is easier than that for P E , ~ ( U E ) . ) Now

uw0;2,1 E,3 (A)'

u WgJJ(X)'.

Moreover, we have (1) W2,',(X)' consists a single class [V], i.e., the one with gr(V) = U&@ A, which contains two vector bundles, i.e., 0; @ X and I2 @ A; (2) WZ,',(X)' U W;,",<x)' U W;';'(X)' N _ P1 with W1;2(X)o consists of 4 classes

[ V ]i.e., , these such that g ( V )= U E @ where denotes any of the four square roots of A. Clearly then in each class [ V ]there , are also two vector bundles and U E @ 1 2 @ X i ; C?E @ (3) WZ,;(X)' u W g J ( X ) ' u WgJJ(X)' = P2\P'. (3.a) W;,;(X)' consists of 9 classes [ V ] i.e., , these [V]with gr(V) = where X i denotes any of the 9 triple roots of A. Moreover, in each [ V ] there , are three bundles, i.e., (Xi)'", X i @ 12 @ X i and 13 @ X i . (3.b) (W2,:(X)'uW$,;(X)')

U (W~,;(X)'UW~,:'(X)')

is isomorphic to E. More-

over, each class [V]in Wo;2y1(X)oconsists of two bundles, i.e., L ( 2 )@ X @ L-2 and I2 @ L @ X @ L P 2 when gr(V) = L ( 2 )@ X @ L P 2 . (One checks that in fact the refined Brill-Noether loci P' and E appeared above are embedded in P2 as degree 2 and 3 regular curves. And hence the intersection should be 6: The intersection points are at [V]with gr(V) = U$' @ X with multi-

plicity 2, and U E @ corresponding to four square roots of X with multiplicity one. That is to say, the intersection actually are supported on W ~ , ' , ( X ) o U W ~ , ~ ( X ) O . So it would be very interesting in general to study the intersections of the refined Brill-Noether loci as well.)


259

From this analysis, we conclude that

PE,3(X) = ( ( q 2

1 - 1)(q2 - q)(q - 1)

+ (q

-

l - 1) ) l)q(q

As a direct consequence, we have the following

Proposition. W i t h the same notation as above, we have

Also we would reminder the reader that here in fact 2- and 3- torsion points are involved naturally in the calculation.

4.2. Why Use only Semi-stable Bundles 4.2.1. Degree 0 At the first glance, Theorem 2.1.2 and Proposition 3.2.2 suggest that in the definition of non-abelian zeta functions we should consider all vector bundles, just as what happens in the theory of automorphic L-functions. However, we here use an example with r = 2 to indicate the opposite. Thus we first introduce a new zeta function 0 > d2. Then ho(E,V ) = d,; (ii) d l = 0 > d 2 . Here two subcases. (a) L1 = O E , then ho(E,V) = 1; (b) LI # OE, then h o ( V )= 0; (iii) 0 > d l > d 2 . Here ho(V) = 0. Thus note that the support of ho(E,V )is only on the cases (i) and (ii.a), we see that similarly as before, the contribution of strictly positive degree rank 2 bundles which are not semi-stable to the zeta function is given by

262

LIN WENG

Hence, we have

-

N? (4-

Q

( q t - 1)(q2 - 1 )

Nl +-.-

1

q - 1 qt - 1 '

I hope now the reader is fully convinced that our definition of non-abelian zeta function by using moduli space of semi-stable bundles is much better than that of others: Not only our semi-stable zeta functions have much neat structure, we also have well-behavior geometric and hence arithmetic spaces ready to use. In a certain sense, these non-abelian zeta functions are quite similar to the so-called new forms: Only after removing these not-semi-stable contributions, we can see the intrinsic beautiful structures. For more details, please see [We2]. REFERENCES

[A] E. Artin, Quadratische Korper im Gebiete der hoheren Kongruenzen, I,II, Math. Zeit, 19 153-246 (1924) (See also Collected Papers, pp. 1-94, Addison-Wesley 1965) [At] M. Atiyah, Vector Bundles over an elliptic curve, Proc. LMS, VZI, 414-452 (1957) (See also Collected works, Vol. 1, pp. 105-143, Oxford Science Publications, 1988) [DR] U.V. Desale & S. Ramanan, Poincarb polynomials of the variety of stable bundles, Math. Ann 216,233-244 (1975) [HN] G. Harder & M.S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles over curves, Math Ann. 212,(1975) 215-248 [HI H. Hasse, Mathematische Abhandlungen, Walter de Gruyter, Berlin-New York, 1975. [Mu] D. Mumford, Geometric Invariant Theory, Springer-Verlag, Berlin (1965) [W] A. Weil, Sur les courbes alge'briques et les varie'te's qui s'en de'duisent, Herman, Paris (1948) [Well L. Weng, Stability, and New Zeta Functions for Number Fields, to appear [We21 L. Weng, A Program for Geometric Arithmetic, preprint, 2001

LIST OF PARTICIPANTS Nakayama, Hiroto Nakayama, Noboru Namikawa, Yoshinori NguyenKhac Viet Nishiyama, Ken-ichi Oguiso, Keiji Ohbuchi, Akira Pearlstein, Gregory J Shimada, Ichiro Sumihiro, Hideyasu Sun, Xiaotao Takahashi, Takeshi Takeda, Yoshifumi Tokunaga, Hiroo Tsuda, Noriko Usui, Sampei Weng, Lin Yamada, Shin-ichiro Yanase, Daisuke Yoshhara, Hisao Yoshioka, Kota Zhang, De-Qi

Ambro, Florin Aoki, Masao Ashikaga, Tadashi Chen, Jungkai Alfred Choi, Jaeyoo Choi, Youngook F'ujisawa, Taro Fujita, Taka0 Goto, Yoshihiro Hanamura, Masaki Harui, Takeshi Hashimoto, Toshiyuki Hwang, Jun-Muk Ikeda, Atsushi Inaba, Michiaki Ishii, Akira Ishii, Shihoko Ito, Yukari Izumi, Shuzo Kawaguchi, Shu Kawahara, Tomokazu Kawanoe, Hiraku Keem, Changho Kei, Miura Keum, JongHae Kim, Sung-Ock Kitagawa, Shinnya Kobayashi, Masanori Kojima, Hideo Konno, Kazuhiro Lee, Seunghun Lee, Yongnam Lu, Steven Shin-Yi Miura, Kei Miyaoka, Yoichi Moriwaki, Atsushi Mukai, Shigeru Murakami, Masaaki

263

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ALGEBRAIC GEOMETRY

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Algebraic Geometry, la Rabida, Spain 1981: Proceedings

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