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Xis etale and hence Xt is a surface of the same type as X (in fact, KXl = 0 and q(Xl)>2, hence by table I, q(Xt)=2). We can deduce quickly that <j) and hence <j>t are all finite morphisms : in fact, if not, let £ c l b e a curve such that <j)(E) is a point ee Alb X. Then considering the Stein factorization X—->7—>Alb of 0, we see that E can be blown down in a birational map X—>7, hence (£ 2 ) 1 are connected.
Enriques' Classification of Surfaces in Char. /?, II
41
for each /, l~l(e) consists of /4 points et 6 Alb X, and 0~1(^) contains a curve Ei that is contracted by t. These curves are disjoint since t (Et) =ei=£ej=l (Ej). Thus Xt has /4 disjoint curves Et with (£?)l*. But for all surfaces of the same type as X, B2=6. This is a contradiction i f / > 1, hence is finite. Next,fixZo, an ample sheaf on Alb X. It follows that Lt=^>t (A>) is ample on X,, with Hilbert polynomial independent of/. By the Main Theorem of Matsusaka-Mumford [7], there is also a number N independent of / such that LfN is very ample for all /. Therefore the infinite set of k-varieties Xl can all be embedded in a fixed PM with fixed degree. Since there are only finitely many A>varieties of this degree (as k is finite), it follows that all the pairs (Xh LfN) are isomorphic to finitely many of them ! Now consider the facts— a) for any variety X and ample sheaf Z, the group of automorphisms f of X such that/*Z is numerically equivalent to L is an algebraic group ; esp. it has only finitely many components (Matsusaka [6]), b) The group Al of translations by points of order / acts on Xt since by definition, it is the fibre product Xx A i b (Alb, /) ; moreover each g € At carries Lt into a sheaf algebraically equivalent to Lt. Let (Xt, LfN) be isomorphic to infinitely many other (Xv, LpNYs. Then Av acts on Xi. Let Gi(zAut(Xi) be the group of automorphisms / s u c h that/*/,* is numerically equivalent to Lt. Then AvczGi which implies that the order of GL is infinite, hence G°t (the connected component) is positive dimensional. But if G°t contains a non-trivial linear subgroup, then when this acts on Xh it would follow that Xt was a ruled surface : since KXl = 0, this is absurd. Therefore G°t is an abelian variety. On the other hand, Alf=(Z/lfZy, and subgroups of fixed bounded index in Av are inside G°. Therefore dim G?>2. It follows that XL consists in only one orbit under G?, hence Xl is a coset space G?/7/, hence Xt itself is an abelian variety. Finally X itself is now caught in the middle between 2 abelian varieties : >Alb X. With a suitable origin, Xt —>Alb X is then a homomorphism, hence if K is its kernel, we find : Alx^Ai - {(x,x+k)\xeXhkeK}. But Xtx xXtdXiX AihXXt and XLxxXi is (i) £tale over Xu and (ii) the graph of an equivalence relation on Xt. (i) implies that X%XxXv = {(x,x+k) \xeXt,ke K>) f for some subset K'aK, and (ii) implies that K is a subgroup. It follows that X=XijK'y hence X is also an abelian variety.
42
E. Bombieri and D. Mumford
References
[ 1 ] Bagnera, G. and De Franchis, M. : Le nombre p de Picard pour les surfaces hyperelliptiques, Rend. Circ. Mat. Palermo 30 (1910). [ 2 ] Deligne, P. and Rapoport, M. : Les schemas de modules de courbes elliptiques, in Springer Lecture Notes 349 (1973). [ 3 ] Enriques, F. and Severi, F.: Memoire sur les surfaces hyperelliptiques, Acta Math. 32 (1909). [ 4 ] Kodaira, K. : On the structure of compact complex analytic surfaces I, Amer. J. Math. 86 (1964). [ 5 ] Lang, S. : Elliptic Functions, Addison-Wesley, 1973. [ 6 ] Matsusaka, T. : Polarized varieties and fields of moduli, Amer. J. Math. 80 (1958). [ 7 ] Matsusaka, T. and Mumford, D. : Two fundamental theorems on deformations of polarized varieties, Amer. J. Math. 86 (1964). [ 8 ] Mumford, D. : Geometric Invariant Theory, Springer-Verlag, 1965. : Lectures on curves on surfaces, Princeton Univ. Press, 1966. [ 9] [10] : Enriques' classification of surfaces I, in Global Analysis, Princeton Univ. Press, 1969. [11] : Pathologies III, Amer. J. Math. 89 (1967). [12] : Abelian Varieties, Tata Studies in Math., Oxford Univ. Press, 1970. [13] Raynaud, M. : Specialisation du foncteur de Picard, Publ. Math. IHES 38, p. 27. [14] Safarevic, Let al : Algebraic Surfaces, Proc. Steklov Inst. Math. 75 (1965). [15] Tate, J. : Genus change in purely inseparable extensions of function fields, Proc. AMS 3 (1952), p. 400. [16] : Endomorphisms of abelian varieties over finite fields, Inv. Math. 2 (1966).
Scuola Normale Superiore, Pisa Harvard University (Received January 14, 1976)
Classification of Hilbert Modular Surfaces
F. Hirzebruch and D. Zagier I.
Introduction and Statement of Results
1. 1 In the paper [6] non-singular models for the Hilbert modular surfaces were constructed. In [9] it was investigated how these algebraic surfaces fit into the Enriques-Kodaira rough classification of surfaces ([11], [12]). But this was only done for the surfaces Y(p) belonging to a real quadratic field of prime discriminant. We shall solve the corresponding problem for real quadratic fields of arbitrary discriminant. We shall use the notation of [6] and [9] and refer to these papers very often. 1. 2 Let K be the real quadratic field of discriminant D and o its ring of integers. The Hilbert modular group G=SL2{o)l {1, — 1} acts on $Q x $ where Q is the upper half plane. The complex space Stf-jG can be compactified by finitely many cusps. This gives a compact normal complex space of dimension 2 denoted by Stf-jG which has finitely many singularities (resulting from the cusps and the elliptic fixed points of G). If one resolves these singularities in the canonical minimal way, one gets a non-singular algebraic surface Y(D). Thus for any discriminant D of a real quadratic field (i. e. D= 1 mod 4 or D = 0 mod 4, where Z)^>5 and D or Z)/4 respectively is square free) an algebraic surface Y(D) is defined. (Here we have changed the notation of [6] § 4. 5. There Y(D) was called Y(d) where d is the square free part of D.) 1. 3 The rough classification of algebraic surfaces without exceptional curves was recalled in [9] (Chap. I, Theorem ROC). Since the surface Y(D) is regular (see [1] Part I, [2] or [9] Prop. II. 4), it is either rational or admits a unique minimal model which is a Resurface, an honestly elliptic surface (fibred over the projective line) or a surface of general type. Thus there are four distinct possibilities, and we wish to decide for every D which of these four cases happens. It was proved recently that Y{D) is simply-connected ([17] and A. Kas, unpublished). Therefore, the Enriques surface (which is an honestly elliptic surface) cannot occur as minimal model of any Y(D) and the class (rational, blown-up K3 surface, blown-up honestly elliptic surface, general type) of Y(D) can be characterized by the Kodaira dimension A; (7(2))) (defined as the maximal dimension of the images of Y(D)
44
F. Hirzebruch and D. Zagier
under the pluricanonical mappings). Thus Y(D) is rational if and only if K(Y (D)) = — 1, if and only if K(Y(D)) = 0, a blown-up K3 surface a blown-up honestly elliptic surface if and only if ic(Y (D)) = 1, of general type if and only if K(Y(D)) = 2. In Chap. II we shall recall the formulas for the arithmetic genus of Y(D). Since Y(D) is regular, we have x(Y(D)) = l+pg^l. It is easy to see that x(Y(D)) tends to oo for D—>oo and certain estimates (Chap. IV) and explicit calculations will show that (1)
x{Y(D)) = 1 0. Therefore, Fmin(Z>) cannot be a ^3-surface or an honestly elliptic surface, because for such a surface c2l=K2=0. Since c2(Y(D)) tends to oo for Z)—>oo, we can reduce the classification to a finite list. This requires certain estimates. In Chap. IV we will prove :
Theorem 1. IfD>285, then c2(Y(D))>0. (The proof depends on computer calculations.) There are exactly 50 discriminants with c2(Y(D))= 12, then sign ($ 2 /G)=0 if and only if D is not divisible by a prime = 3 mod 4. For Z)=12 the signature vanishes ([6] § 3. 9). As Hammond showed (see 1. 5), the signature of Q2jG vanishes (D=\2 again excluded) if and only if the actions of G on $ 2 and on £ X $~ are equivalent, one direction of this equivalence being clear by the second formula of (13). 2. 3 A "cusp" is described by a pair (Af, V) where M is a complete Z-module in the real quadratic field K and F a subgroup of finite index in the (infinite cyclic) group of all totally positive units e with eM=M. To such a pair (Af, V) we associate in a topological way ([6] § 3) a rational number V) using the continued fraction describing the resolution of the "cusp singularity" of type (Af, V) : There exists a totally positive number a in K such that aM = Zwo+Z-l, where w0 is reduced, i.e. 0 < w0 < 1 < w0. Here x \-> x' denotes the non-trivial automorphism of K. The number w0 has a purely periodic continued fraction development
where ((bOy...9 br_x)) is the primitive period which (up to cyclic permutations) depends only on the strict equivalence class of the module and conversely determines this strict equivalence class. (We recall that by definition the modules M and M are strictly equivalent if and only if there exists an element /3 of K of positive norm such that M=fiM. They are called equivalent if there exists an element /3 of K such that M=pM.) We define (17)
()
Z( O i=0
and (18) l(M) = r. Thus l(M) is the length of the period which we shall also call the length of the module. If y is an element of K with negative norm, then d(rM) = 38(M) = In particular, 8(M)=0 if there exists a unit s of K with negative norm such that v
;
sM=M. To prove (19) we observe that w0 (see above) admits an ordinary continued fraction (20)
i
l
1
Wo = co+ — + ±
(c^z, c^\
which is not necessarily purely periodic. We denote the shortest period of even length by
for
t>0)
50
F. Hirzebruch and D. Zagier
(21) («!,-• •,l). Then 5(a~2, U2) = 2d(cr2) 8(cr\ U2)= d(a-2)=0 /(a" 2, U2) =-2l(a-2) l(a-\U2) = /(a-2)
if N(e0) = 1, if N(e0) = - 1 , if N(e0) = 1, if
Classification of Hilbert Modular Surfaces
51
The numbers d and / depend only on the strict module class. Therefore, 8 and / can be regarded as functions on C+. For the total parabolic contribution we have in view of (16) and (25)
a> = 4a€C - S W a ) ) , (Sq:C->C+) \
(26)
which because of (11) is a relation between continued fractions and class numbers of imaginary quadratic fields. (Compare [6] 3. 10 (55)). The pair (Af, V) determines a singularity whose minimal resolution is cyclic. The number of curves in this resolution equals l(M, V) (see [6] 2. 5 Theorem). The Hilbert modular surface $2/G for the field K of discriminant D is compactified by h points. They are singularities in the compactification ffi/G which when resolved minimally give rise to h cycles of curves. The number of all these curves will be denoted by lo(D). We have El(Sq(a))
27
if Neo= - 1
£
or equivalently (28)
k{D) = 2 l(Sq(a))
(Sg: C+^C+).
a€C+
The Hilbert modular surface {&X{Q~)/G is also compactified by h points. These cusps are of type (ya~2, U2) where y is an element of A" of negative norm. We denote by IQ (D) the number of curves needed to resolve all these cusp singularities minimally. Then
and by (19) and (26) (29)
lo(D)-lo(D) = I2w.
2. 4 Let Y(D) be the surface obtained from $2/G by minimal resolutions of all the singular points (see Chap. I). If we assume Z)^13, we have only quotient singularities of order 2 or 3. Those of order 2 are resolved by one curve ; those of order 3 by one or two curves depending on whether the type is (3 ; 1, 1) or (3 ; 1, —1). As in [9] (Proposition II. 2 and (7)) we conclude e(Y(D)) =e(&IG)+a2(G)+a3+(G)+2a;(G)+l0(D) for D ^ 13. Noether's formula states that c2l{Y(D))+e(Y(D)) = l2x(Y(D)). (12), (29), (30) we obtain
Using (9), (10),
52
F. Hirzebruch and D. Zagier
If we consider the action of G on § X £>~ instead of $ X £>, then '» = ^vpkKP'q)
with
C(/>>9) = 1
of which , i V ^ 2 a n d i ) ^ N,4N, thenF = F^N)/{\,
-1}
and a(F) = N-l. If D even, tf = 2, D * 8, thenT = r i (2)/{l, - 1 } = r o (2)/{l, -1} (16) and(/(r) - 2 . If D = Nor D = 4N(N± 2), thenT = rf(N)/{l9 -1} IfZ) = 8, ^ = The group F{(N) consists of those matrices in SL2(Z) which are of the form + L J modulo N and Ff(N) is an extension of index 2 of F^N). The proof of (15) and (16) is carried out by applying the method of [6] p. 270 to equation (14). To bring F into the above form one must conjugate in GL2(K). Using (9), (15), (16) (and the Proposition in 3. 3 for the case (D/N) = l) we get P r o p o s i t i o n . If N is an odd prime, then the curve FN* has 2l~l components, except in the case D = N or D=4N where it has 2 or 4 components respectively. If N = 2, then FA has 2l~l components if D is odd or if D = 8. If D is even (D=fc8) then F± has 2l~2 components. Remark. For Nj(D the skew-hermitian curves (14) all belong to the same component of FN* and GmjG operates transitively on the set of components. If N\D (iV=£2), then two skew-hermitian curves (14) belong to the same component if and only if the two values of (r/N) are both equal to + 1 or both equal to — 1. In [7] § 3 it was stated that the curve FN((N/p)^ — l) on Y(p) (p prime) is irreducible. This has to be corrected as pointed out by Hammond. It will be shown in a forthcoming dissertation by Hans-Georg Franke (Bonn) that FN is irreducible if N^O (p2). \i N=0 (p2) then FN has exactly two components. 3. 7 An exceptional curve on an algebraic surface is a non-singular rational curve of self-intersection number — 1. If the surface is regular and not rational, then any two exceptional curves are disjoint and can be blown down simultaneously. In this section we assume that Y{D) is not rational. Thus we exclude 10
discriminants (Chap. I (1)). How many exceptional curves can be found on Y(D) using skew-hermitian curves ? For a discrete subgroup F of PL2 (R) with §/F of finite volume the number (17)
cY(F) =
ix() r«>2
was introduced in [6] 4. 3. We recall that e denotes the Euler number, ar[F) the number of /^-equivalence classes of fixed points of order r of F and a(F) the number of cusps. If a component E of a skew-hermitian curve in Y(D) has the
Classification of Hilbert Modular Surfaces
61
non-singular model £ / / \ then where cx*E denotes the value of the first Chern class of Y(D) on E. Since Y(D) is not rational, c^E^l implies that c^E=l and E is an exceptional curve (see [6] 4. 4 Corollary I). The curve F{ has 2L~l components (see the Proposition in 3. 3). Each component passes through a quotient singularity of order 2 and one of order 3 on ffi/G and is on Y(D) an exceptional curve which gives rise to a configuration
-3
-2
(18) -1
of non singular rational curves. These configurations are disjoint to each other. Using FY we have found on Y(D) in this way 3-2'" 1 curves which can be blown down.
The groups F which occur for the components of F2, F3 are Fo(2) and Fo(3) respectively (to be divided by {1, —1}). For the components of F4 we have F = T0(4) or F=F'{2) if D is odd (to be divided by {1, —1}). (These groups were treated in [6] 5. 5 if D is a prime.) If D is even, the group for F± is JT 0 (2)/{1, —1}. If 3\D (Z)^ 12), then the components of F9 have the group Fl(3)=F0(3) (to be divided by {1, - 1 } ) . For these groups F (namely To(2), To(3), T'(2), T0(4), always divided by {1, —1}) the value of c{(F) equals 1. Since Y(D) is supposed to be not rational, all components of F2, F3, FA and (if 3|Z>) F9 give exceptional curves. Each
component of F2 passes through a quotient singularity of order 2 on ) = n(l+ z < (_2)) + i - n ( l + Z l ( - 3 ) )
blow-downs. (Note that &( —iV) = (sign Dt) Xi(N)-) Again we conjecture that the surface Y°_(D) obtained by these blow-downs is minimal. IV.
Estimates of the Numerical Invariants
4. 1 The purpose of this chapter is to prove the facts x(Y(D)) = 1 o D = 55 8, 12, 13, 17, 21, 24, 28, 33, 60, c\(Y(D)) < 0 =>Z> c\(Y.(D)) D< 136, c\{JT(p)) < 0 => p < 821
(p= l(mod 4) prime)
(compare Chapter I), thus reducing the problem of classifying all Hilbert modular surfaces to the consideration of a finite list. Since all of the invariants have been calculated (by computer) up to at least Z)=1500, it will suffice to prove (1) (2)
D > 1500=>z(7(Z>)) > 1, x(Y.(D)) > 1, c]{Y{D)) > 0, c\(Y.(D)) > 0, />>1500=>*?(y r (£))>0.
There are precisely 50 discriminants for which the four inequalities of (1) are not all satisfied ; complete numerical data on these discriminants is given in section 4.5. 4. 2 As explained in 2. 1, the dominant term in the formula for all of these numerical invariants is
w!)
*2=Z>(mod4)
From ox{n)>n-\-\ we deduce easily that (4) C K (-l)
64
F. Hirzebruch and D. Zagier
this result can also be obtained by writing
and applying the functional equation of £K(s). From 2. 4 (31) and 2. 1 (9), (10), (12) we have c\{Y{D)) = 4 C * ( - l ) - , 7 - / T ^
l
and
moreover, by 2. 3 (29),
and hence
Z)3/2+?(7(i)))
> ^ j f ^ +1
(D > 20°)-
Hence the inequality x(F(Z)))>l in (1) will follow once we have proved that c\(Y{D)) is positive ; since (by 2. 2 (15) and 2. 4 (33)) the values of x and c\ for F_(-D) are at least as large as for Y(D)> the remaining two inequalities in (1) will also follow. Thus to prove (1) we have to show (5)
4C K (-l)-/ 0 -( J D)-a 3 + (G)/3>0
(Z)>1500),
while for (2) the inequality
(6)
2C / f (-l)-|/ 0 (/>)-|£ + ^ > 0
(/>
will certainly suffice (equation 2. 5 (40)). We introduce a new invariant (7)
l(D) = S+/(a)
(notation as in 2. 3). IfD—p is a prime, then Sq : C+—>C+ is an isomorphism and if, however, D has t distinct prime factors, then Sq has a kernel of order 2c~l and so l (9) l~(D) = 2 KrSq(a)) = 22'-' The advantage of working with l(D) rather than l0 (D) is that it can be evaluated by a formula analogous to formula (3) for C,K( — 1). Indeed, l(D) is the sum of the lengths of all cycles occurring as the primitive period of the continued fraction of some quadratic irrationality w of discriminant D (the discriminant of w is de-
Classification of Hilbert Modular Surfaces
65
fined as b2—4ac, where aw2-\-bw+c=0, (a, b, c) = l). This is simply the number of reduced quadratic irrationalities w of discriminant D (i. e. w satisfying zt>> l>w' >0), since, as discussed in 2. 3, such w have purely periodic continued fractions, and a cycle ((#o,---,£r-i)) of length r gives rise to precisely r reduced numbers
If aw2-\-bw-\-c—0, b2—4ac=D, then the condition (a, b, c) = l is automatically satisfied since D is the discriminant of a quadratic field. Therefore
= #{(«, *, c) eZ> | ^ - 4 a c = A The inequalities are equivalent to a>0, \-b-2a\) > ^y-y/D-200
since the right-hand side is > - o j
-3.6
^—for Z)>1500, this implies (6).
By (3) and (10), the left-hand side of (11) equals
(•2) with
(D > 730) ;
66
F. Hirzebruch and D. Zagier
We have —0.6 ,, v ^ n—14 3 Q
(13) v
y
for n < 50, ~ -^ for n > 50.
Indeed, for w 5 6 ^ = ( 7-~-j>
-ar[(-4H'-4)+T S ('+7-1')] Kd0). For Z)>729 there are at most 4 values of £ (two positive and two negative) for which k=D(mod 2) and 0 t 23,000 or t = 5 and and D> 60,000 or t = 6 and D > 157,000 or t = 7 and D > 420,000.
(17)
But the smallest discriminant with t=l is 4. 3. 5. 7. 11. 13. 17= 1,021,020>420,000, so (17) implies that (16) holds for all D with t=l. A similarargument holds for any t>7, since a D with t>l distinct prime factors is greater than 4. J. 5. /. 11. U. 1b
- g 5 5 3 6 (^
J,
so 2^2420,000. Therefore (17) implies that (16) (and hence (14)) holds for all D> 157,000, and a computer calculation showed that (14) holds for all D up to this point. 4. 5 As already stated, the calculation of the various invariants for Z)W of degree 2 onto a P l -bundle Plover J, which induces a double covering C-^>Pl. This rational map, in turn, determines its branch locus B on W. By applying "elementary transformations" to W, we normalize the singularities of B. By this process, W is transformed into another P'-bundle Wl over J, which turns out to be the image of the rational map 0 associated with |A^+^*m| for sufficiently ample m. If J = P\ Wi and 0 can be identified respectively with W and / as defined at the beginning. In the final § 5 we shall make a remark on pencils of curves of genus 2 with base points. 1) Supported in part by NSF grant MPS72 05055 A02.
80
E. Horikawa
Corresponding results are valid over any algebraically closed field of characteristic ^ 2 . This paper is an outgrowth of afterthoughts about [3], § 2. In fact, the whole paper depends only on several lemmas there and other more or less standard results. In addition, the surfaces of type II studied in [3] serve as clarifying examples for the present study. 1.
A Fundamental Construction
Let g : S—>J be a surjective holomorphic map of a compact complex analytic surface S onto a non-singular complete algebraic curve A of genus n. We assume that a general fibre of g is an irreducible non-singular curve of genus 2. By results of Kodaira [5], this implies that S is projective algebraic. We also assume that S contains no exceptional curve (of the first kind) in any fibre of g. Let K denote the canonical bundle of S. For any divisor m on J, we set fl(m) = dim H°(S, 0(K+g*m)). Let p be a point on J. Then £(m, p) =a(rt\) —a(m—p) is lower semi-continuous in p. Hence, if/? is a generic point of J, b(m, p) does not depend on the choice of p, and we denote it by b(m). L e m m a 1. For sufficiently ample m, we have fl(m) = pa+% deg HT+TT,
where pa denotes the arithmetic genus of S.
Proof. By the Riemann-Roch theorem and the Serre duality, we have aim) = / > a + l + d e g m + d i m / / 1 ( S , 0(-£*m)). Hence it suffices to prove dim// J (5, 0{-g*m))
=degm-l+;r
for sufficiently ample m. We assume that xn = ^pi is a sum of distinct points and i
that Ci=g'l(pi) are irreducible and non-singular. In view of the exact sequence 0 -» G(-g*m) -» 0 -> 0 0Ci — 0, i
we need to prove dim Ker(W(S, Os)^®H\Ct,
0Ct)) = it.
For this purpose we use the exact sequence 0 -> H> (J, 0A) -> W (S, 0s) -+ H°(J, R>g.Os). This implies that the above kernel is at least ^-dimensional. Furthermore we have H°(Jy (Rlg*Os) (X) 0(—m)) = 0 for sufficiently ample m. It follows that any element of H^S, Os) which vanishes on S C t is in the image of H\A, GA). The following lemma is an immediate consequence of Lemma 1.
Pencils of Genus 2
81
L e m m a 2. For sufficiently ample m, we have i(tn)=2. Let/? be a "variable" point on A and C=g~l(p). We let ^ denote the restriction to C of the canonical bundle K of 5. If C is non-singular, Kc coincides with the canonical bundle of C. Let @j denote the set of those divisors m on A such that £(m)^l, and let nti be a minimal element in &{. Then we take a divisor m2 such that the images of H°(S, OiK+g^m,)) and H°(S9 G{K+g*m2)) generate H°(C, 0(Kc)) for any general fibre C. Such a divisor exists by Lemma 2. We further assume that m2 is minimal among those which have the above property. We set b=m2—TUi. By our choice of nti and m2, we can find C* € #°(£, O(K+.g*mt))9 i=l9 2, such that their restrictions to C form a basis of H°(C, 0(Kc)). The pair (£„ C2) defines a generically surjective rational map of degree 2. Here P(0{—nti)®0( — m2)) denotes the P ! -bundle over J associated with 0 ( - m 1 ) © 0 ( - m 2 ) . This P'-bundle is identified with P(0(b)©0) and we shall denote it by Ib. Let Zf : £—>$ be a composition of a finite number of quadric transformations such t h a t / o XS extends to a holomorphic m a p / : £ —>2'b. We let q : Zb-^>A denote the natural projection and take a section Ao of ^ whose normal bundle is [—b] via an obvious identification. This section will be called the 0-section of Ib. Let R denote the ramification divisor of/, i.e., i f / i s given locally by (zi, £2)—>(^i5 w2), i? is the divisor of the zeroes of the Jacobian determinant d(wu w2)ld(zi, Z^}- We define the branch locus B off to be the direct image/,/? of R as a divisor. We note that B has no multiple component. L e m m a 3. There exists a line bundle F on Ib such that 2F=[B]9 the line bundle associated with the divisor B.
where [B] denotes
Proof. Clearly B is linearly equivalent to 6A0^-q*0 with some divisor 6 on A. Let Z be a very ample divisor on Sb such that q induces a biholomorphic map Z—> A. We assume that Z intersects B transversally at a finite number of points. Then Z=f~l(Z) is an irreducible non-singular curve and the intersection cycle BZ coincides with the branch locus of Z—>Z. From this we infer that the line bundle [6] is divisible by 2. q. e. d. Let F be a line bundle on Ib which satisfies 2F= [B]. We take a covering {£/J of Ib by sufficiently small open subsets and assume that F is defined by a system of transition functions {fj} on the nerve of the covering {[/*}. Then we can take the equations ^ = 0 of B on Ut which satisfy bi=ftjbj on Ut f] Uj. Let wt denote fibre coordinates on F over Ut. We define a subvariety S' ofF by the equations w2i = bi over Ui% We call S' the double covering of Ib in F with branch locus B. Since B has no multiple component, S' is a normal complex space. We note that Sf depends on the choice of F.
82
E. Horikawa
L e m m a 4. We can find a line bundle F on Ib satisfying 2F= [B] such that there exists a birational holomorphic map from S onto the double covering S' of2b in F with branch locus B.
Proof, We set W=2h. Let $-*S'-+W be the Stein decomposition o f / a n d let / ' : S'—>W be the induced holomorphic map. Then S' is normal and f*6§=J\0'S" Since/'is flat ([2], IV, (6. 1. 5)),/; is invertible. Hence we obtain an exact sequence
with some line bundle F on W. Let {Ut} be a covering of W by sufficiently small open sets Ut and let {1, Wi\ be a basis off'*6s> over each Ut. We may assume that w\ is in F(UU Gw) and we denote it by bt. Then the branch locus B is defined by the equations bi=0. It follows that F satisfies 2F=[B] and that S' is nothing but the double covering of W in F with branch locus B. q. e. d. 2.
Elementary Transformations
Let S' denote the double covering of Ib in F with branch locus B as in Lemma 4. In this section we shall apply elementary transformations to 2b in order to study the singularities of B. We set W=2b. Let x be a singular point of B of multiplicity m. Here the multiplicity means the sum of the multiplicities of irreducible components of B at x. Let p : WX—>W be the quadric transformation with center at x and let L=p~1(x). We set Bx=p*B — 2[m/2]L where [m/2] denotes the greatest integer not exceeding m/2. Let Sx be the double covering of Wx in Fl=p*F— [m/2] [L] with branch locus B{. Then we have a natural birational holomorphic map S{—>S' which is compatible with p (see [3], § 2). Let M be the proper transform of the fibre of W—*A through x. Since M is an exceptional curve on Wl9 we can contract M into a non-singular point. Let q : W^,—•W^ be the contraction of M and set B2=q*Bx. We let n denote the multiplicity of B2 aty=q(M). In view of the equality MBl = 6—2[m/2], n is equal to 7 —2[m/2] or 6 —2[m/2] according as M is a component of Bx or not. In either case we have B{ = q*B2—2[n/2]M. Let F2 be the unique line bundle over W2 which satisfies q*F2=Fl-\-[n/2] [M], Then F2 satisfies 2F2=[B2]. Therefore we can consider the double covering S2 of W2 in F2 with branch locus B2. Again we have a birational holomorphic map SY—>S2 which is compatible with q. Thus, from the birational point of view, we can replace the triple (W, B, F) by (W29 B2, F2) as above. It is well known that W2 is a P ] -bundle over J. We shall call this operation the elementary transformation at x. L e m m a 5. After a finite number of elementary transformations, the triple (Ib, 5 , F) is transformed into another triple ( W , B\ F') such that B' has no singular point of multiplicity
Pencils of Genus 2
83
greater than 3.
Proof. If B has a singular point x of multiplicity ^ 4, we apply the elementary transformation at x. Retaining our previous notation, jy is of multiplicity ^ 3 on B2. Therefore our assertion follows from the theorem on resolution of singularities of curves on surfaces. q. e. d. In view of [3], Lemma 5, we need to study "infinitely near triple points" of B'. Let sx be a triple point of B' and let W[-^W be the quadric transformation with center at sx. If the proper transform B[ of B' has a triple point over sx, let us call it s2. We note that s2 is unique if it exists. Next let W'2-^W[ be the quadric transformation with center at s2, and look for a triple point over s2 on the proper transform of B[. We continue this process until we obtain a maximal sequence sx, % * • -j> *k of triple points over sx. In this situation we call sx a k-fold triple point. If k= 1, we call it a sirhple triple pointy while if A;J>2, we refer to the collection (jl5 s2, •••, J,) as infinitely near triple points. L e m m a 6. After a finite number of elementary transformations, the triple (W\ B\ F') is transformed into another triple (Wl, B*, Ff) such that B* has only the following singu-
larities (0), (I,), (II,), (III,), (IV,) and (V) with jfc^l. (0) A double point or a simple triple point. (I,) A fibre F plus two triple points on it {hence these are quadruple points of Bx). Each of these triple points is (2A:— 1) or 2k-fold. (II,) Two triple points on a fibre, each of which is 2k or (2k +1)-fold. (Ill,) A fibre F plus a (4£—2) or (4k—l)-fold triple point on it which has a contact of order 6 with F. (IV,) A 4A; or (4k-\-1)-fold triple point x which has a contact of order 6 with the fibre through x. (V) Afibre.F plus a quadruple point x on F, which, after a quadric transformation with center at x, results in a double point on the proper transform of F.
Proof Let x be an m-fold triple point of B' with m^>2 and let F be the fibre of W—>d through x. Let/? : W[—*W be the quadric transformation with center at x9 L=p-X(x), Bi=p*B' — 2L and let M be the proper transform of F. By the assumption m^>2, the proper transform B^ of B' has an (m— l)-fold triple point xx on L. Let q : Wx-+W2 be the contraction of M and B2=q^B{. We first assume that F is a component of B'. Then C=B'—F has a double point at x and the proper transform C, of C has a double point at the intersection of L and M. Since we have Bl = Cl-JrL-\-M, B2 contains q(L) as a component. Moreover C2=q^Cl has a quadruple point at q(M). Thus B2 has a singularity of type (V). Next we shall study the case in which F is not a component of B'. This case will be divided into two subcases according as x{ is on M or not. First assume that xx is on M. This implies that B' has a contact of order 6 with F at x. Hence, if m=0 or 1 mod 4, B' itself has a singularity of type (IV,) with A;=[m/4]. While, if m=2 or 3 mod 4, B2 contains q(L) as a component and B°2=
84
E. Horikawa
B2—q(L) has an m-fold triple point x2 on q{L). Furthermore, since B°l^=Bl—L has a contact of order 3 with L at #l5 B°2 has a contact of order 6 with q(L) at x2. Thus Z?2 has a singularity of type (III*) with k=\mj^]. Next we assume that x{ is not on M. This implies that 5 ' has a contact of order 3 with F at x. If B' has another triple point on F, let us call i t j . We may assume thatjy is w-fold with nW* denotes the natural projection.
Proof We note that S is a double covering of a surface W in a line bundle F with a branch locus 6. Here Ti^ is obtained from W% by a finite number of quadric transformations. Let n : V—> W be the completion of F as a i^-bundle. n admits two sections Wo and W^ such that [ Wo] = [ W^] -f TT*P. As a divisor of V, S is linearly equivalent to 2 Wo. Let L denote the pull-back on W of the line bundle [JJ]+^*I. We use the following exact sequences on V: 0 -> 0 ( T T * Z - ^ ) -> 0(T:*L) -+ 0&{n*L) -+ 0,
0 -+ 0 ( T T * L - ^ ) -> 0(n*L- Wo) -> Ow{L-F) -> 0, 0->r*L- rt^) ->0(w*Z) -
The last two yield an isomorphism In view of the first exact sequence, it is sufficient for our purpose to prove H°(Wy Ow{L-F))=0. But this follows from f ( Z , - F ) = - 2 < 0 , where f denotes a fibre of the projection H^—>J. This completes the proof of Lemma 12, hence that of Lemma 11. We now prove the last assertion of Theorem 1. If A=P\ we may assume that W*=2bi. Then we have, by Lemma 11, fl(m) = dim//°(J, 0 ( f - c + m ) ) + d i m / / ° ( J , 0 ( f - c + m - b * ) ) . This implies that b is linearly equivalent to b1 or — b*. Hence W% can be identified with 2b. q. e. d. Lemma 11, combined with (1), has as a consequence the following theorem.
Pencils of Genus 2
89
T h e o r e m 2, With the same notations as above, the geometric genus pg and the irregularity q of S are given by q = ;r + dim Our next purpose is to study the relation between S and S. We note that S contains the exceptional curves Ea and E'a, a = l , 3, 5, •••, over each singular fibre (see Lemma 8). L e m m a 13. S is obtained from S by contracting the exceptional curves Ea and E'a with odd a.
Proof First we note the following fact. Let E be an exceptional curve on S which is contained in a fibre of g : S—>rf. Then E is a fixed component of |2C+g*m| for any
m. Let S be the surface obtained from S by contracting Ea and E'a with odd a and let g : .S5—>J be the natural projection. In order to prove our assertion, it suffices to show that there is no exceptional curve in any fibre of g (see [8]). But, as we have noted above, such an exceptional curve is the image of a fixed component of |i?+g*m|. Hence it is the image of a component of ^71 (see the proof of Lemma 11). This is impossible by Lemmas 8 and 10. q. e. d. This leads us to the following Theorem 3. We have
where v(*) denotes the number of singular fibres of type Proof By [3], Lemma 6, we have
Combining this with (1), we obtain K2=2pa—4-f 671. Now the assertion is an immediate consequence of Lemma 13. q. e. d. We conclude this section with the following theorem. T h e o r e m 4. Assume that K is ample. Then the branch locus Bx has no singularities except those of type (Ij).
Proof From Lemma 8 and the above observations, it follows that a singular fibre of any other type (including that of type (0)) results in a non-singular rational curve with self-intersection —2. q. e. d. 5.
A Remark on Base Points
In this section we consider the case in which S contains an exceptional curve E
90
E. Horikawa
which is mapped onto J(hence we have 7r = 0). In other words, if S—>S0 is the contraction of £, So carries a pencil of curves of genus 2 with a base point. As far as algebraic surfaces of general type are concerned, we have the following T h e o r e m 5. Let S be a minimal algebraic surface of general type which contains a linear pencil \C\ of curves of genus 2 with C 2 >0. Then S has the following numerical characters : pg^2, q=0 and c\=\.
Proof We borrow an argument from Kodaira [6]. Let K denote the canonical bundle of S. Then we have KC+C2=2. Since both KC and C2 are positive, they are equal to 1. Hence we have C(K— C)— 0. Then, by Hodge's index theorem, (K— C)2=K2— 1 is non-positive. Since K2 is positive, we conclude that K2=\. This implies that q=0 andj&^^2 ([1], Theorems 9 and 11 or [6]). q. e. d. The surfaces with pg = 2, q=0 and c\=\ are studied by Kodaira in [6]. They admit holomorphic maps of degree 2 onto a quadric cone in P 3 . The inverse images of lines through the vertex of the cone form a pencil of curves of genus 2 which has one base point (see [4], for details). References [ 1 ] Bombieri, E. : Canonical models of surfaces of general type, Publ. Math. IHES 42 (1973), 171-219. [ 2 ] Grothendieck, A. : Element de Geometrie Algebriques, Publ. Math. IHES 4, 8, •••I960 ff. [ 3 ] Horikawa, E. : On deformations of quintic surfaces, Invent. Math., 31 (1975), 43-85. [4] : Algebraic surfaces of general type with small c\, II, Invent. Math., 37(1976),121-155. [ 5 ] Kodaira, K. : On compact complex analytic surfaces I, Ann. of Math. 71 (1960), 111-152. [6] : Pluricanonical systems on algebraic surfaces of general type, II, unpublished. [ 7 ] Namikawa, Y. : Studies on degeneration, in Classification of Algebraic Varieties and Compact Complex Manifolds. Lecture Notes in Math. 412, Springer, Berlin, 1974. [ 8 ] Shafarevich, I. R. : Lectures on Minimal Models and Birational Transformations of Two Dimensional Schemes, Tata Inst. Fund. Res., Bombay, 1966.
Department of Mathematics University of Tokyo (Received November 5, 1975)
New Surfaces with No Meromorphic Functions, II
M. Inoue 0 0.
Introduction
In our previous note [3], we have constructed a kind of (compact complex) surface S satisfying (a)
b{(S) = 1,
b2(S) ± 0
and (/3)
S is minimal.
As for the significance of these conditions we refer to Kodaira [4], [5] or [2], [3]. In this note, we shall construct other such surfaces and study their properties. They have close connections with the cusps of Hilbert modular surfaces. We shall use some methods devised by Hirzebruch in [1] to resolve the cusp singularities of Hilbert modular surfaces. In particular, we shall refer to [1] for some results on continued fractions and real quadratic number fields. Notations. As usual, we denote by Z, Q, R, R+, Cand C* the ring of rational integers, the field of rational numbers, the field of real numbers, the set of positive real numbers, the field of complex numbers and the multiplicative group of nonzero complex numbers, respectively. We denote by [—oo, oo], [0, oo] and [—oo, 0] the intervals which are mapped homeomorphically onto [—1, 1], [0, 1] and 2 [—1, 0], respectively, under the principal value of —tan"1.
1.
Algebraic Preliminaries
For a real quadratic irrational number x, we denote by x' the conjugate of x, by M(x) the Z-module generated by 1 and x> and by Q(x) the real quadratic number field obtained by adjoining x to Q. Let U(x) = {aeQ(x) | a > 0 , a*M(x) = M(x)}9 U+(x) = {ae U(x) | a-a > 0}. 1) Supported in part by the Sakkokai Foundation.
92
M. Inoue
Then U(x) and U+(x) are infinite cyclic groups and [U(x) : U+(x)] = 1 or 2. # has a unique development as an infinite (modified) continued fraction which is periodic from a certain point on : where ^ e Z , et>2 for f > l and ^ > 3 for at least one j>t. x is purely periodic (namely, /=0) if and only if # > l > # ' > 0 (see [1]). Throughout this note we shall fix a real quadratic irrational number w satisfying co > 1 > a)' > 0. a) is developed uniquely into a purely periodic continued fraction : 0) = [[«o, »i,-», «r_,]] (1) where r is the smallest period and 02, rij > 3 for at least onej. Let Mx and M2 be .Z-modules of rank 2 in Q(co). We say that Mx and Af2 are strictly equivalent if there exists deQ(co) such that 0, (T>0 and d»Mx = M2. For the following two lemmas we refer to § 2 of [1]. L e m m a 1. For any Z-module M of rank 2 in Q( 1 > T)' > 0. L e m m a 2. Let ^,, rj2 be irrational numbers in Q((o) satisfying the condition (2), and let be the development of r]x as a purely periodic continued fraction. Then M{r)x) and M(y2) are strictly equivalent if and only if for some teZ,
00. ii) U+((o) = U+( then ^y1 belongs to U+((o). iv) Let (3)
* as a purely periodic continued fraction, where s is the smallest period. If there exist 7], y eQ(w) such that ^ > 0 , / < ( ) , y • M (co) = M (y) and 37 > l > ) / > 0 , then Y]=[[mh mt+l, •••, mt+s.i]]for some te Zy 00, e'*, 3eQ{oj) such that0, 0. We set /3 = 3-s. Then /3>0 and P - M ( ( o ) = d-M = M ( o > * ) ,
i8
/
= ( d - e ) ' = d'*ef
< 0,
q. e. d.
We take o>* and /3 as in Proposition (1. 1). We take a generator a of the infinite cyclic group U+(w) such that a > l , and define integral matrices N, N* and B by (a., l).N=a.(0
/ i8
and a;* —(co*)'>0, we
< 0.
+
ii) If [U(w) : C/ (o>)] = 2, there exists aoe U((o) such that a o > 1 and a^* and a0 as /3. Then we have nt = mt for all z, r = s, ^ = N*,B2 = N. For any integers i, k we define ^ = rij, m k = mh
i =j (modulo r), k = / (modulo J ) ,
0 <j < r—1, 0 < / < s— 1,
94
M. Inoue
where nh ml are the natural numbers in (1), (3). Then we know that for any i, k, (7) ni9 mk > 2 (8) n^ mL> 3 for at least one j , I. We define a)^ w% e Q(w) inductively by
Then wiy o>^>0 and a)i = Q)i+r for a n y i,
w% = CD*+S for a n y k.
W e set
Co*)"1
{wx 1 G)Q*G)_X
for i > 1 for i = 0
c^i+i
for
ft>*)~1
(<w,* 1 &>*#<jt>*i
^ ^ — 1
5
for A: > 1 for A: = 0
^?+i
for
AJ< — 1 .
Then (9) implies
_
°k-2> °k-
For a proof of the following lemma, we refer to (2. 5) of [1]. L e m m a 3.
I/a = ar = bs.
Proposition (1. 2).
L-i,orL-i,oj '" L—i, ojProof. From (10) it follows that \nr_u
L
By Lemma 3 and the definition of #i? we obtain ar_x=o)\a,
ar=l/a9
)=2y for yeL. Hence n*\L is injective, and moreover we have from (3. 6) j 2 ). q. e. d. Lemma 3. 3. n* (HY) = n* (L) 0 F. Proof It suffices to show that TT^(L) and F are orthogonal to each other. Take xl=K*(yl) (y}eL) and x2=n:^(y2) (y2eFl). From (3. 7) and (3. 8), we see that 7r*(xl)=2yl eL and x*(x2)=y2Jrc*(y2) eFx@F2. Hence we have 2(*,.*2) = {n*xx.n*x2) = (2j 1 .(j 2 +^(j 2 ))) = 0.
q.e.d.
L e m m a 3. 4. Let M' be the minimal primitive sublattice of Hx containing M. Then
det AT = |det M^| = 26. Proof Obviously (M')± = M±, and |detM'| = |det {M'Y\, because M' is primitive in the unimodular lattice Hx. Since d e t M = d e t (&v&fl)=28, we have only to prove that the index [Mf : M] = 2. Let 0v be the homology class of the curve &v. 8
Take an element x=^av6v(aveQ) of M\ x$M. Since x»6veZ, av must be halfy= l integers. We may assume av — 0 or 1/2. Then the number m of av with av=l/2 is either 4 or 8, because x2=—m/2 is an even integer. Assume m = 4 and
4
x=^0vl2.
If we take a divisor Z) in the homology class x (D2=x2=—2), the Riemann-Roch theorem implies that D or —D is linearly equivalent to an effective divisor D'. But D~D', since 2D~£ev.
As D'6v=x8v= — \ for l < v < 4 , Z>' contains
as components : D'= i^&v+D\
D">0. It follows that 2D"~ — 2J@y? which is a
l
l
8
contradiction. Therefore m = 8 and # = 2 0 y / 2 . This shows [M r : M ] < 2 . Con8
versely, the element2#v/2 belongs to M', because we have l
in which Z)^ (^=1, 2) is the sum of irreducible components with multiplicity > 2 in the singular fibre @~l(tk) of type I6* or IV*. q. e. d. T:*(TY) = Tx. Lemma 3. 5. n*{HY) = ML, Proof For the first equality, we know that the inclusion n*(HY) ciM1- holds. By Lemmas 3. 3 and 3. 2, we have
128
T. Shioda and H. Inose
|det n+(HY)\ = |det x*{L)\ = 26 |det L\ = 26, since F and L are unimodular. Then 7r*(HY)=M± by Lemma 3. 4. Next we shall show that 7i*(TY)(zTX:) i.e. if ye TY, then (TT+(J>) •#)=() for all ^ 6 ^ = 7 ^ . Indeed this is obvious if x e M. If x e ML fl^x? then TT*(X) is defined and belongs to SY. Therefore, by (3. 6), we have 2{x,(j>).x) = {K*K*{y).**{x)) = (2yn*{x)) = 0. This proves 7r^(TY)aTx. Similarly we have 7t*(Tx)(zTY. Now, by Lemma 3. 3, we have (noting M(zSx) Tx = SxczM-= n,(HY) = K^L) 0 T. But Tx is orthogonal to FaSx, and hence TxCW to the first factor, has the 3 singular fibres of type IV* : y/r
1 o v f 4-3F
(i—\ 2 31
and the 3 sections Gl5 G2 and G3. Applying (1.2), we see that ^ ( 7 ) ^ 2 0 and r(W)=0. Then, by (1. 3), we have det TY = 33/n2 with n = n{W)>3. Hence det TY \2 1" q. e. d. = 3, and £^F~ 1 2• L e m m a 5. 2. Let A = Ctx Ciy Ct being the elliptic curve with the fundamental periods 1 and i=y/—\. Let a be the automorphism of A defined by o(z\, Zi) = (iZ\, —iz2)- Then the minimal non-singular model Y of the quotient surface A/(a) is a singular K3 surface such
Proof Since a2 is the inversion cA of A, it has 16 fixed points. Among these, the 4 points (vu Vj) (l 2 in the singular fibre W~x{s^) of type II* (cf. (3. 4)). Since the involution c of Y acts trivially on L (cf. Lemma 3. 2), we deduce the following isomorphism : (6. 15) Hy/Sy =i Hy/Sf ff HXjSX. As in (I), this implies (6. 16) P2{Y mod p, T) = P2(X mod p, T) for almost all p, and hence we have (6.17) H{YIK,s) = 1i{XIK,s). This completes the proof of Theorem 6.
q. e. d.
Remark. Theorem 6 and its Corollary were known for some special singular K3 surfaces, e.g. the Fermat quartic surface defined over Q{e2*m) (Weil [11], [12], cf. [9] p. 105) and the elliptic modular surface of level 4 defined over Q(\/—l) (Shioda [7] p. 57). The former corresponds to the matrix L
J and the latter to
4 0
References [ 1 ] Deuring, M. : Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins, Nachr. Akad. Wiss. Gottingen (1953), 85-94. [ 2 ] Inose, H. : On certain Kummer surfaces which can be realized as non-singular quartic surfaces in P\ J. Fac. Sci. Univ. Tokyo, Sec. IA, 23 (1976), 545-560. [ 3 ] Kodaira, K. : On compact analytic surfaces I I - I I I , Ann. of Math. 77 (1963), 563-626 ; 78 (1963), 1-40. [ 4 ] Pjateckii-Sapiro, I. I. and Safarevic, I. R. : A Torelli theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR, 35 (1971), 530-572. [ 5 ] Safarevic, I. R. : Le theoreme de Torelli pour les surfaces algebriques de type K3, Actes,
136
T. Shioda and H. Inose
Congres intern, math. (1970) Tome 1, 413-417. [ 6 ] Safarevic, I. R. et al. : Algebraic surfaces, Proc. Steklov Inst. Math., 75 (1965) ; AMS translation (1967). [ 7 ] Shioda, T. : On elliptic modular surfaces, J. Math. Soc. Japan, 24 (1972), 20-59. [ 8 ] Shioda, T. and Mitani, N. : Singular abelian surfaces and binary quadratic forms, in "Classification of algebraic varieties and compact complex manifolds", Springer Lecture Notes, No. 412 (1974), 259-287. [ 9 ] Tate, J . : Algebraic cycles and poles of zeta functions, in "Arithmetical Algebraic Geometry", Harper and Row (1965), 93-110. [10] Vinberg, E. B. : Some arithmetical discrete groups in Lobacevskii spaces, in "Discrete subgroups of Lie groups and applications to Moduli", Tata-Oxford (1975), 323-348. [11] Weil, A.: Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc, 55 (1949), 497-508. [12] Weil, A. : Jacobi sums as "Grossencharaktere", Trans. Amer. Math. Soc, 73 (1952), 487495. [13] Weil, A. : Basic number theory, Springer (1967).
Department of Mathematics University of Tokyo (Received November 7, 1975)
On the Minimality of Certain Hilbert Modular Surfaces
G. van der Geer and A. Van de Ven° 1.
Introduction
For some time now Hirzebruch and others have studied certain fields of Hilbert modular functions from a geometric point of view (see [2], [3], [5]). This leads to the introduction of a non-singular algebraic surface Y°(p) for all square-free positive integers /?. In [5] the question is settled how the surfaces Y°(p) fit into the rough classification of algebraic surfaces, at least for those values of/? which are prime and congruent 1 mod 4.2) It turns out that for/?=5, 13 and 17 the surface Y°(p) is rational, that for/?=29, 37 and 41 this surface is an elliptic X3-surface, that for/? = 53, 61 and 73 it is a minimal honestly elliptic surface, and that for/?> 89 the surface Y°(p) is of general type. As already follows from this description, the surfaces Y°(p) are minimal (i.e. without exceptional curves of the first kind) for 2989 one has the following conditions for exceptional curves E on Y\p) :
(a)
En*(E)=t;
(b) if Cis a non-singular rational curve with C2=— 2, —3 or —4, then CE = 89, the minimality of Y°(p) follows already from these simple rules. However, in other cases this is not sufficient. Sometimes (like for/? = 193) a very special argument gives the desired result, but most effective seems to be a reasoning used many times in the cases /? = 229 and/? = 293, which in its simplest form can be described in the following way. Suppose Y°(p) is not minimal. Then, by (a), there exist at least two disjoint exceptional curves on Y°(p). Let Y be the surface obtained from Y°(p) by blowing down these two curves. We havep g (Y) =pg(Y°(/?)). On the (hypothetical) surface Y we construct a particular positive canonical divisor K== J^atAu
a,e Z, a, > 1,
the A^s being irreducible curves on Y (the divisor K is chosen differently in different situations). We then single out one of the curves Ai9 say Ao, to be the "final curve". By intersecting K with the other curves At we obtain inequalities of the following type at > (XiOo + Pi,
aiy & e Q , at> 0 .
Finally, we consider the intersection number and find it to be more than KA0, a contradiction. To illustrate this, we describe a simple example. Let Z b e a configuration formed by three curves : Ao, Ax and A2. The curve Ao is a rational curve with one ordinary double point/*. The curves Ax and A2 are nonsingular rational curves. Both Ax and A2 have only the point/? in common with Ao. Al=—3 (hence KA0=3), A2x=A22=—2, AOAX=AOA2^2, AXA2=\. complex surface X a configuration JC exists, then pg{X)2. Then there exists a positive canonical divisor K on X, passing through/?. Such a divisor K contains Ao, Ax and A2, and hence it can be written as K = aoAo+axAx + a2
with tf0? a\-> #2>1) RA0>0, RAx>0,
RA2>0.
Intersection with Ax gives
2ao-2ax+a2+RAx = 0 2ao—2ax + a2 < 0,
and similarly intersecting with A2 gives 2ao+ax-2a2
X be a holomorphic involution on X, such that the quotient Xjc is a non-singular rational surface Y. If Kx is the canonical divisor class on X} then c induces the identity on \KX\.
Proof. Let R be the ramification curve on X. By a well-known result of Hurwitz we have Kx = a*(KY)+R, where a : X-+Y is the canonical projection. From this formula it follows in our case that there cannot be a divisor De\Kx\, containing R, for a*(D—R) would be a divisor in |2A"r|, which is impossible, because Y is rational. Now let s be any section in the canonical bundle of X, different from the zero section. By the preceding remark, the restriction sR of s to R does not vanish identically. The holomorphic function C*(SB)SR1 is equal to a constant c, c^O. The section e*(s)—cs of the canonical bundle vanishes on R and hence vanishes identically on X. So we find c*(s)=cs, and c*(s) and s have the same zero divisor. This proves the proposition. The following remark also will turn out to be quite useful for the cases p=229 By a chain of curves on a complex surface we mean a set of irreducible curves Cu •••, Cn9 with the following intersection behaviour : n n
f *•
**
^ * ~~ I 0
17 ^1 =
*
otherwise.
Proposition 2. 9. Let Bl9 •••, Bn be a chain of'( — 2)-curves on the surface X, and let Al9 •••, Am be irreducible curves on X, all of them different from Bu •••, Bn. Suppose AiBl = AiBn=ai^0for
f = l , ••-, m. If f^atAt + ^bjBj+R i=\
with AtR>0,
BjR>0
is a canon-
j=\
ical divisor on X, then j
for allj=l9
—,n.
Proof. If for this canonical divisor all at vanish, then the statement is trivial. Otherwise, we have bj>l for all j = 1, •••, n. Now suppose bj>k for all j=l9 •••,« m
and l
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G. van der Geer and A. Van de Ven
But this and .7 = 1
implies b2>k+l, etc. Thus the proposition follows by induction with respect to k,
Finally, a word or two about the diagrams. These diagrams don't show always the whole picture, that is, curves may have more points in common than shown in the diagram. For example, in the diagram for p = 229, the curve Fp passes through the intersection point of T3 and T4 (which also lies on So). The intersection properties (and other information) used in the diagrams are generally to be found in [3] and [5]. However, § 3 of [4] is needed to see that on 7°(293) the curves i%3, F3l and F31 are intersected by different curves Tu and that different curves 0t meet Fl7 and F37. From [5], p. 20 it follows that Fp (the ramification curve on Y°(p)) is not a rational curve, and hence not an exceptional curve if/>>89. 3.
The Main Result
Theorem 3. 1. The surface Y°(p) is minimal for 89 -^ a+1. Intersecting Z) again with F9 wefind—2d ^-a+2^-a+l
Next, X5 8 =2 yields us b>-ja+-j.
All
this information, together with Proposition 2. 9 and the fact that Fl5 is contained in R, gives a contradiction if we intersect D with So. For case 4) the argument is very similar. In case 5) we take after blowing down, a canonical divisor aSB+bS9+R, 1, b>l, RS8>0, RS9>0, and find by successive intersection with S8 and S9 that a>b and that — 3A + 2J + -o-(fl + i) + -n"A+l\, RS8, •••, RS'9>0, which contains £I3, F n , F25 and B2. Intersection with 5*9 gives
+
^
+ < 1,
b>a.
Then intersection with S8 yields ^ ^
\ < 2, i.e. «, (d) it is sufficient to exclude the following cases : ES0>3 ES0 = 2, EO = 1, where 0 is one of the curves 0,, •••, 0 8 ES0 = 2, ET = 1, where T is one of the curves 71,, •••, T8 £5 0 = 1, EO = ET = 1, T and 0 as in 2) or 3)
On the Minimality of Certain Hilbert Modular Surfaces
147
5) ES0 = 1, ET = ET' = 1, where T and 7" are among Tl9 T39 T5 and T7, T^ T'. For the cases 1) and 2) it is sufficient to intersect (after blowing down E and c{E)) the curve So with a canonical divisor of the type aS0+R, a>l, RS0>0, to obtain a contradiction. To exclude case 3), we blow down E and e(E), obtaining KS0— 4, and consider a canonical divisor of the type aS0+bFl7-\-cF37-{-R, a, b, c>l, RS0>0, RFl7>0, RF37>0. This divisor contains T, e(T), T9 and TIO all at least a times, and Sx and S[ at least a + b+c times (Proposition 2. 9). Hence we would find :
-Aa+2a+Aa+2b + 2a+2b + 2c < 8, again a contradiction. In case 4), after blowing down six times, a canonical divisor of the type aS0-{bF37+R leads to the usual contradiction. The last case, case 5), requires a little bit more care. After blowing down 2? and c(E) we have KS0=—Sl=l0. The curve S0 intersects three ( — 2)-crosses, namely the pair T, T' ; the pair c(T), c{T') and the pair T9, Tl0. We claim that it is possible to find a canonical divisor D of the form with #,-•-, A> 1, and RS0, •••, Rc(T')>0. In fact, the divisors in the at least 4-dimensional linear system of canonical curves, which pass through the intersection point of T and T\ through four general points of So and through two general points of Fl7, all contain So and Fl7 at least once and £„ S\ at least twice (Proposition 2. 9). The linear subsystem, passing through two general points of .F37 contains F37, and hence S2 and S2 at least three times. So there is at least one canonical divisor in this subsystem passing through two general points of F3l9 hence containing F3l, and this is a divisor of the type mentioned above. Application of Proposition 2. 9 gives/^4, and subsequent intersection with £,, S2, Fl7, Sx and S2 gives £>4, f>5, b>2, e>5, / > 6 . Since F3l, F37 and i%3 intersect different T/s, we may assume that T and e(T) meet one of the curves F3l, F37 or ^43. If T and c(T) meet F3l or JF37, then g>2, h>2, and intersection with So yields a>2. If T and c(T) meet JF43, then intersection with F43 shows that D contains i%3, and as before we conclude that a>2. Proposition 2. 9 implies/>« + 6 + rf+l ; intersection with Sx and ^2 now gives e>a-\-b-\-d-\-l. Intersection of D with F37 and Fl7 leads to
-6d+d+2(a + b + d+l)+2a < 8, — + —i and
148
G. van der Geer and A. Van de Ven
Finally, intersection with So yields ft ft
\ and 1, 3£ coincides with X. Then we have the following theorems.
160
K. Akao
T h e o r e m 2. Suppose that X satisfies the above conditions, and that p>\ and q>\. Then there exists a finite holomorphic map from X onto a submanifold X' of a Calabi-Eckmann manifold V such that the following diagram commutes : X
>X'
>V
I W\X'
W ^
W
>
# ' ( * - ( * ) , Z)^Ht(*-*(p), 0,-Hpy)^H>(x-(n-i(p), Z), where both rows are exact. Note that j? : Hl(X, Ox)^>Hx{n-x{p), 0n-iiP)) is an isomorphism, as can be proved by using the Leray spectral sequence in § 1. Since clearly j*tf*A and j*7z*L2 are trivial line bundles, j?a~ln*Lx and j?a-l7z*L2 lie in the image of i*. Since they are linearly independent over JJ, they generate a sub-lattice of i*Hl(n~l(p), Z). But, since n~l(p) is a non-singular elliptic curve, i*Hl(7t-l(p),Z) is a lattice in Hl(ic'l(p)9 0^(P))^Hl{X, Ox)=Hl{X, Of), hence TT*Z,J and TT*Z,2 generate a lattice in Hl(X,
0jjf).
q- e. d.
Proof of Theorem 2. First we choose coordinate coverings {£/>} of X and {M^} of M^such that ic(U$) = Wt. Let laAj be the transition functions of the line bundle La ( a = l , 2) with respect to {W^\ . Then, by Lemma 2 and Lemma 3, we can find non-vanishing holomorphic functions pi and qi on each U% a holomorphic function Xfj on each U^f] U) (=£0), and a complex number £ with Im c^O such that the following equations are satisfied :
where we put / a ,iy=l for e=/. Since La is very ample, we can find a basis {£$} (A=0, 1, -.., -AT) of i/°( W, O(La)) which gives an imbedding j a of W7 into PN% where iVa=dimci/°(M/r, 0(La)) — 1 ( a = l , 2). Then by the above equations, we can construct a holomorphic map from X into F=(C A l + 1 —(0)) X (CN2+l — (0))/G, which qi**Z$l> ~; qi**W)]. Here G= {£} is the maps *€ £/? to [(^7r*CK, - , Pfr*^\ one parameter group defined in § 2, and for z € CNl+l — (0) X C^2+1 — (0), [>] denotes the image of £ in V. The manifold Fis nothing but a Calabi-Eckmann manifold and we get the following commutative diagram: X
>V
i
On the other hand, every positive-dimensional closed subvariety of V is of the form (p~l{Y) for a closed subvariety 7of PN* X PNi ([3]). Therefore (p : X-*p(X) is a finite map. For a general point/* of IV, the elliptic curve n~x(p)=E is a finite unramified covering of Q)^H*(PpxPq, Q) was already proved in Corollary 4 in § 1. Now we have only to show that the image J by TT of the set of the singular fibres of it: X^W has only simple normal crossings for sin-
162
K. Akao
gularities. By Holmann [5], the action of E admits a holomorphic slice at every point of X. Let p e W be a singular point of J, and Fp the stabilizer subgroup of E at a point of TC~1 (p). Then there exist an open polydisc neighbourhood U of p in W, and a holomorphic slice U at a point of n~l(p), which is stable under Fp and also isomorphic to an (n—l)-dimensional polydisc. Moreover UjFp~ U via rc, and the image by n of the set of fixed points of Fp in U coincides with AC\U. Since FpczE, Fp is a finite abelian group. Therefore we choose a coordinate system Z={Zi, ", Zn-\) on I? such that every element y^Fp is expressed by a diagonal matrix with respect to these coordinates. Let Uf be another (n—l)-dimensional polydisc and w=(wl, •••, wn_x) be a coordinate system on U\ Put g=#|/ 7 />|, and consider the map xs : #—>£/' defined by tyfo, •••, £n_i) = (£f, •••, *£_,). Then tf factors through 7r|# :
and Tf'(A)CL {(w) e U/\wlw2'"Wn_l = 0}. Hence J has only a normal crossing at/?. The smoothness of irreducible components of J can also be proved easily. q. e. d. Proof of Theorem 3. In this case 7r,(Z)^Z 5 SO q{X) = l, and bl(W)=q(W)=0. Since Wis protective, nx(W) is a finite cyclic group. Let Wbe the universal covering of W. Then by Theorem 1, X—XxwW admits a holomorphic torus action. But Hence H*(W)^H*(P«) by Corollary 4 in § 1, and bv(W)\ for 0} (*=0, 1, ••-, N) be a basis of Z/°(W, 0(L)), where A^-l=dim//°(W/ 0 (L)). This gives an imbedding of Winto P^. Then, as in Theorem 2, we can construct a holomorphic map from X to F=C^ + 1 — (0)/(a n ) neZ which maps A; € U% to [ ( ^ T T * ^ , •••,^7r*^iNr))]. F i s a (homogeneous) Hopf manifold, and the rest of the proof is the same as that of Theorem 2. q. e. d. Remark. If X is homeomorphic to Sl X S5, then we can get rid of the assumption of the flatness of n : X^>W. Proof Since n^W) is finite, we may assume obviously that nl{W)= {1}. Then, by Theorem 1, X admits a holomorphic action of 1-dimensional complex torus E.
Complex Structures on S2P+l X^ 2« +1 with Algebraic Codimension 1
163
Let
Vbe the resolution of singularities of V. Then, since dim c PK=dim c F=2, (poXS is a composite of quadratic transformations. Now consider the Leray spectral sequence for V^W and Qv ; E*>*=H*(W, R*zr*Qv)=*H*+*(V, Q). We shall show that £° >2 =0, which means that XS is a finite map. But then since W is non-singular and XS is birational, XS is an isomorphism, and TZ is flat. Since XS is birational, the dimension of the support of RXZS^QV is zero, so E22A = 0. Since W is non-singular, b3(W) = bl(W)=0 by the Poincare duality, hence E32>°=0. This implies that E°2'2=E°J. But since PK is projective, 4 2 ( ^ ) > 1 5 a n d from t n e inclusion //^M 7 , Q)d^H2 (F, Q), we have E\*=E™~H2(W, Q). Since A2(K) = 1, we get E°2>2 = 0. q. e. d. References [ 1 ] Akao, K. : On certain complex structures on the product of two odd dimensional spheres. Proc. Japan Acad., 50 (1974), 802-805. [ 2 ] Brieskorn, E. and Van de Ven, A. : Some complex structures on products of homotopy spheres. Topology 7 (1967), 389-393. [ 3 ] Calabi, E. and Eckmann, B. : A class of compact complex manifolds which are not algebraic. Ann. of Math., 58 (1953) , 494-500. [ 4 ] Hironaka, H. : Review of S. Kawai's paper, Math. Rev., 32, #466 (1966), 87-88. [ 5 ] Holmann, H. : Quotientenraume komplexer Mannigfaltigkeiten nach komplexen Lieschen Automorphismengruppen. Math. Ann., 139 (1960), 383-402. [ 6 ] Holmann, H. : Seifertsche Faserraume. Math. Ann., 157 (1964), 138-166. [ 7 ] Iitaka, S. : On algebraic varieties whose universal covering manifolds are complex affine 3-spaces, 1, in Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo (1973), 147-167. [ 8 ] Kato, Ma. : Complex structures on Sl X S5. Proc. Japan Acad. 49 (1973), 575-577. [ 9] : Complex structures on Sl X S5, J. Math. Soc. Japan 28 (1976), 550-576. [10] Kawai, S. : On compact complex analytic manifolds of dimension 3. J. Math. Soc. Japan, 17 (1965), 438-442. [11] Kodaira, K. : On compact analytic surfaces II. Ann. of Math., 77 (1963), 563-626. [12] : On the structure of compact complex analytic surfaces I. Amer. J. Math., 86 (1964), 751-798. [13] Miyaoka, Y. : Kahler Metrics on Elliptic Surfaces. Proc. Japan Acad., 50 (1974), 533-536. [14] Tits, J. : Espaces homogenes complexes compacts. Comment. Math. Helv., 37 (1962-63), 111-120.
Department of Mathematics Gakushuin University (Received December 15, 1975)
Defining Equations for Certain Types of Polarized Varieties
T. Fujita Introduction
In this paper we improve a result of Mumford [5]. To be explicit, we fix our notation and terminology0. Every variety is assumed to be defined over an algebraically closed field K. For line bundles L, Af on a variety V we denote by R(L, M) the kernel of the natural multiplication homomorphism r(L)®r(M)-+r(L-\-M). A line bundle I o n V is said to be simply generated if r{tL)®r(L)-+r({t+\)L) is surjective for every t^> 1. L is said to be quadratically presented if'it is simply generated and if the natural homomorphism R(sL, tL)(x)r (L)—>R(sL, (t-{-l)L) is surjective for all s,t^l2). Now we state the following T h e o r e m (Mumford). Let L be a line bundle on a smooth curve C of genus g. Then L is simply generated if deg L^>2g+1, and L is quadratically presented if deg L}>3g-\-l.
We improve the above result in the following three ways. First, we can weaken the assumption that C is smooth. Second, we show that L is quadratically presented if deg L^>2g+2. Third, in the complex case, we give a higher dimensional version of these results, an embedding theorem and a structure theorem for certain types of polarized varieties, which will play an important role in our study of polarized varieties (see [la]). As an example of applications, we give in § 5 a criterion characterizing smooth hypercubics. 1.
One-dimensional Case
Throughout this section, let C be an irreducible reduced curve with hl(C, 0) =g. We remark that C is locally Macaulay, i.e., 0x is a Macaulay local ring for any xeC. Hence there is a canonical dualizing sheaf w on C (see [2]). 1) Basically we employ the same notation as in [1], oo
2) L is simply generated if and only if the graded 7C-algebra G(V, L) — @ P(V, tL) is generated by r(V,L). Then L is quadratically presented if and only if all the relations among the elements of F(V, L) in G(V, L) are derived from quadratic ones.
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T. Fujita
Definition 1. 1. A coherent sheafs on C is said to be quasi-invertible if it is torsion-free and of rank one. Remark 1. 2. co is quasi-invertible. Any non-trivial subsheaf of a quasi-invertible sheaf is also quasi-invertible. A sheaf is quasi-invertible if and only if it is locally isomorphic to a non-zero ideal of Qc. Definition 1. 3. For a quasi-invertible sheaf J, we define deg J and A(J) as follows :
L e m m a 1. 4. Let J'i ~§ be quasi-invertible sheaves and let
F(D, LD) is surjective. Hence Df)Bs\L\ = and Bs \L\aBs \L—D\. This proves a), b). We have the following commutative diagram :
o _> r(M-D) 0_>
(g) r(L) -> r(M) 0 r(L) -> r(D, MD) ® r(L) -^ 0 ->
r(M+L-D)
r(M+L)
-•
r{D,[M+L]D)
0 0 0 It is easy to see that the rows and the first and the third columns are exact. Hence the second column is also exact, proving b). c). By R(M, LD) we denote Ker (r(D, LD) (g) r(C, M)->r{D, [M+L]D)). Then we have the following commutative diagram : 0
0
1 R(M,L-D)
->
r{M) (g) r(L-D) 0—
r(M+L-D)
0
1 R(M,L)
9 ->
1 R(M,LD)
— r(M) r(Z) -> r(M) ® r(D, LD) -> 0 —
r(M+L)
-+ r{D,[M+L]D).
0 Every sequence above is exact. It follows that ^ is surjective. Next we consider the following diagram :
R(M, L) /?(M, Z^ (g) r(/)) -> 0 72(^,4
>R(M,L+D)
>R(M,[L+D]D).
The dotted arrow is defined by a\-^>a®d, where d e F(D) is the section defining D. Then the above diagram is commutative and the rows are exact. Moreover, (p is surjective since Bs \D\ = . Tracing this we prove c). q. e. d. L e m m a 1. 9. Let L, M be line bundles of degree I, m respectively. Suppose that m^>2g, and L is general. Then p : r ( Z ) ( g ) r ( M ) - > r ( Z + M ) is surjective. Proof In view of Lemma 1. 8. b), it suffices to consider the case l=g+l, by
168
T. Fujita
replacing L by L—D, D being a general effective divisor of degree l—g—1. In this case, we have hl(L)=0, h°(L)=2 and#y \L\ = . LeU,, 12 be a basis for H°(L). Then {^ = 0} f] {^=0} =^. Therefore, ^ 2 = > ^ i € P(L+M) for F(D, LD) is surjective. A sequence V=Dn'z>Dn_xZ)--'D Dx of subvarieties of Fwith dim Dj=j is called a ladder of {V, L), if each Dj is a rung of (-Dy+i, L). A ladder is said to be regular (resp. smooth) if each rung of it is so. Proposition 2. 2. Let D be a rung of a prepolarized variety (V, L) defined by 8 e F(V, L). Let f a ( a = l , - - - , k) be homogeneous elements of the graded algebra G(V, L) = ®F(V,
tL) with deg £a = da, and let rja be the restriction of fa to G{D, L) = ©T(D,
tL)7\
Suppose that {ya}a=i,-,k generate the algebra G(D, L). Then G(V, L) is generated by 8 and -»aJ a =
\,-,k'
Proof. Let A be the graded subalgebra of G( V, L) generated by [8, f,, • • -, gk] and let At = AnT(V, tL). Let pt be the restriction r(V, tL)-+r(D, tL). Then we have Ker pt = 8F(V, (t—\)L). Moreover, F(V, tL)(zAt + Ker pt since G(D,L) is generated by [rja] • Combining them, we obtain F(V, tL)=At by induction on t. C o r o l l a r y 2. 3. Let D be a regular rung of a prepolarized simply generated, then so is L.
variety [V, L). If LD is
P r o p o s i t i o n 2. 4. Let {V, L), D, 3, {fa} and {rja} be as in Proposition 2. 2. Let gj (lfgjG(D, L) and n : Q-^R be the ^-algebra homomorphisms defined by q(X0) =8, ? ( X , ) = ^ rij,)^, x(X0)=0 and x(Xj) = Yj for l^j^k. Letting ft : (£-+d be the multiplication by Xo, we have the following commutative diagram : 0
1
K e r q —>
i /< d i
G(V,L)^
K e r r
1 —•
R
->0
i
- ^ G(Z>, L) - » 0
I
0
All the columns and the middle and the bottom rows of this diagram are exact. Hence so is the top row. Therefore, there is/ a € Ker q such that n{fa) =ga for each l:gaF, a linear system A on V with Bs A''= and an effective divisor E on V such that 7i*A=E-{-A'. Moreover, W=pA,(V')l0) is determined by (F, A) and is independent of the choice of (V, it, A', E).
Remark 3. 2. A general member of A is irreducible if dim W^2 and if codim Bs A^2 (see, for example, [1], p. 114). Definition 3. 3. A is said to be non-degenerate if dim W=dim V. Proposition 3. 4. Let (V, L) be a prepolarized variety and suppose that V is locally Macaulay, dim Bs \L\ ;g0 and that \L\ is non-degenerate. Then (F, L) has a ladder.
Proof Remark 3. 2 says that a general member D of \L\ is irreducible. Moreover, D is locally Macaulay and generically smooth, hence D is reduced. So D is a rung of (F, L). Repeating this process we obtain a ladder. q. e. d. Proposition 3. 5. Let (M, L) be a prepolarized manifold such that dim Bs \L\ fgO, d(M, L)^2d(M, L) — lU). If\L\ is degenerate, then a general member of\L\ is smooth.
Proof. Since every base point of \L\ is isolated, it suffices to show the existence of a member D of \L\ which is smooth at any given point peBs \L\. Assume that 9) Recently the author proved similar results also for the cases in which char A^>0. (1976, Oct.) 10) pA> denotes the rational mapping associated with A. 11) For a prepolarized variety (V, L) with dim V=n we define (cf. [1]) d(V,L)=Ln and n+d(V9L)-!fi(V,L).
Defining Equations for Certain Types of Polarized Varieties
171
such a member does not exist. Let Ml be the monoidal transform of M with center/^ and Ep the inverse image of p. Then mEp is a fixed part of \L\Mx for some integer m^> 2. Applying Theorem 3. 1 to the pair (Mly AY) where Al = \L\Ml — mEpy we have a manifold M' together with a birational morphism TT : M'—*MU a linear system Af with Bs Af = and an effective divisor E such that ?r* Al = E+Af. Let W=pA,(M') and let / / be the natural hyperplane section on W. Then HM, = [A'] and consequently LHn~l {M'} =L(L—mEp—E)n-l=Ln>0, whererc= dim M. Thus dim W= n—l, since |L| is degenerate. Put w = deg W and let X be a general fiber of pA,. Then £ / ) {Z}>0 since Ln — mn=[Al]n^[Al]Hn-1=LHn-l — mEpHn-l=Ln--mwEpX. Therefore, d=LHnl = (H+E+mEp)Hn-l^mEpHn-l = mwEpX^mw^2w. On the other hand, 0^A{W, H)2A— 1. q. e. d. Corollary 3. 6. Let (M, L) be a prepolarized manifold such that dim Bs\L\^0, d(M, L)^2A(M, L)- 1 and d(M, L)>0. Then (M, L) has a ladder. Proof. We may assume that \L\ is degenerate, since otherwise we can apply Proposition 3. 4. So a general member D of \L\ is smooth. Since dim Bs |L|^0, we have Bs \lL\ = for />0. \IL\ is non-degenerate since d(M> L)>0. Hence Hl(M, —L) = 0 (see Mumford [6]). Therefore D is connected, and is a rung of (M, L). Now we have d(D, L)=d(M, L) and A(D, L)0, A — J(M, L)^g(My L)™=g and that dim Bs |L|^0. Then a) (M, L) has a regular ladder if d^>2A— 1, c) g(My L)=A(M, L) and L is simdly generated if d^> d) L is quadratically presented if d^2A-{-2. Proof a) Corollary 3. 6 implies the existence of a ladder {Dj} with dim D~j. We have d(DlyL)=d and g{D{,L)=g. If this ladder were not regular, then J(Z),, L) \
In general, KV^LKV H°Q* log 25, is an isomorphism.
Proof By definition,/*a> is a rational ^-form. For any point px€ Vl9 choose a local system of regular parameters (wl3 •-, wn) and for p2=f (px) choose a local system of regular parameters (Z\,'-,Zn) such that if p2eD2, Zi'-Zr—Q defines D2 aroundp2 and ifpx e Dx, wx--ws=0 defines Dx around px. By definition of/, f~l(Dx) dD2 and so (L) Zi = Tlw¥i£i} nij ^ O5 where et is non-vanishing around/?,. Thanks to /dL\ dzi _ yrn dwj det £i \ L ) ^ ,~ ~^~ ' we see that/*o> is logarithmic. Now, assume t h a t / i s proper. ThtnfV{ is closed in V2, hence fVx =fVx=V2. Letting j : V2-^f-[Vly we getf=(f\f-lVl):J,_vthich l is proper. Hence j is also proper. Thus V2=f~ Vl. In other words, f~lDx = D2. Furthermore assume/to be birational. Take a logarithmic ^-form wx on Vx along D{. Then we have a rational ^-form o>2 on V2 such that a>1=/*a)2- Since / is birational and Vx is complete, there exists a Zariski open subset V°2 with codim (V2—Vl) ^ 2 such that the inverse g : V2—>VX of / exists,, that is, f*g is the open immersion VQ2—>V2. Since wx is a logarithmic F1? F3 being an algebraic variety, such that/-/^ is a morphism. For example, a dominant rational map from a complete Vx into a non-complete V2 is not strictly rational. A rational map from Vx into a complete F2 is always strictly rational. Actually, consider a compactification Vx of Vx and regard / as a representative of a rational map / : Vl—>V2. Then there is a birational morphism p. from a complete variety F3 onto Vi such that / • / / is a morphism. Put V3=p~lVx. Then {i=p\V3 is proper and / • / i is a morphism. Consider a strictly rational map from an algebraic variety Vx into an algebraic variety V2. As in § 1, taking compactifications V? and F2* of non-singular models Vf—>F15 V}-+V2 of Fi and F2, with smooth boundaries Dx and 5 2 , respectively, we obtain a linear map :
On Logarithmic Kodaira Dimension of Algebraic Varieties
179
/ * : H°Q*logD2^H°Q«(*) U
H°Q« log Dx.
By definition, there is a birational proper morphism n : V3—>VX such that g=f*pt is a morphism. Choose a suitable compactification F3* with smooth boundary A such that p is a morphism and g=f*p is a morphism. By Proposition 1, g * : H°Q H°Qq log 5,. Note that in case/is dominant,/* is injective. Thus, in general, if dim Fj = dim V2 — n, and / i s dominant, then l
0
K,-.,myx
^ pmi,...,mnv2.
In particular, if F, is a non empty Zariski open subset of F2, then PmV^PmV2, KV^KV2 and qV^qV2. We say that a birational m a p / : Vl-^V2 is proper i f / a n d / " 1 are both strictly rational. For example, a proper birational morphism is a proper birational map. Note that V= P 17 P if there is a proper birational map Vl-^V2. Proposition 2. Let V be a non-singular complete algebraic variety and D a divisor of normal crossing type. Assume that K~\-D is ample. Put V= V—D. Then any strictly rational dominant map f: V—+V is a restriction of an automorphism of V.
Proof From the argument above, we infer that
/ * : H°m{R+D) -> H°m{K+D) is an isomorphism. Since m(K-\-D) is a hyperplane section for m>0, we get the conclusion. q. e. d. Example 5. Let D be a divisor of normal crossing type in Pn. Assume that . Then Aut{Pn-D) aPGL{n,k). For a variety V in Pn, we write Lin
V = {a € PGL{n3
k);aV=V],
S Bir V = the group generated by strictly birational maps : V—>V. Then under the conditions of Proposition 2, S Bir V = Lin V, which is a finite group (see Theorem 6).
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S. Iitaka
Let Vx and V2 be non-singular algebraic varieties and / a dominant morphism P^—>F2. Choose compactifications V{ and V2 with smooth boundaries Dx and D2 of Vx and V2> respectively such that the rational map / : Vl-^V2, f\Vl=f is a morphism. Then there is an effective divisor Rf which satisfies
(R)
Rx+Dx~f*{R2+D2)+Rf
Rf is called the logarithmic ramification divisor for f.
We prove (R) by using the notation in the proof of Proposition 1. A rational logarithmic 0 and PmVl = PmV2 for sufficiently large m. Then the rational map @mi,Vl associated to Im^ + D^l is factored as follows :
Proof As in the proof of Theorem 3 in [3], choose mt so large that Om.yV. : Vt—* 0muV.(Vi) is birationally equivalent to the ^ + Z)rcanonical fibered manifold of Vt for i = l , 2 (see [4]). Choose m=a multiple of L. C. M. (ml9 m2) so that PmVi = PmV2. Then the natural inclusions :
H°m(K2+D2) c H°mf*(K2+D2) c H°m(f*(K2+D2)+Rf) = ^ ( m ^ + A ) ) are isomorphisms. From this follows the conclusion. Using this we can prove easily T h e o r e m 1. If KVl = n, PmVl = PmV2for map from Vx into V2 is birational.
q. e. d.
m > 0 , then any dominant strictly rational
As a generalization of a theorem of Peters [10], we prove T h e o r e m 2. Let Vbe a non-singular algebraic variety with icV7>0. Then any dominant morphism from V into itself is an etale covering map from V onto V.
Proof Let V be a compactification with smooth boundary D. f is regarded as
On Logarithmic Kodaira Dimension of Algebraic Varieties
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a representative of a rational map / : V—+V. Performing a finite succession of monoidal transformations with non-singular centers c D on V, we get a birational morphism p : V*—»Fsuch that f*p—g is a morphism F*—+V. Then by (R)
m(R*+D*)
~g*m(K+D)+mRg,
where K* denotes a canonical divisor on V* and D*=p~lD ; and
m(^* + S*) ~ p*m(K+B)+mRft. Hence, With the decomposition f=p*g in mind, we write
Then
m(K+D) ~mf*(K+B)+mRf. By assumption /cF;>0, there is an m>0 such that \m(R+D)\ 3 C(m). Write g*C(m) as a sum of effective divisors D* and 8 where & consists of the components that are mapped to Oby/z*. Similarly write p*p*D*=D*+6'. Note that <S, <S'>0. Then, since
