Resolution of Singularities of Embedded Algebraic Surfaces
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Resolution of Singularities of Embedded Algebraic Surfaces
PURE A N D APPLIED MAT H EMAT I C S A Series of Monographs and Textbooks
Edited by PAULA. SMITH and SAMUEL EILENBERC Columbia University, New York 1 : ARNOLD SOMMERFELD. Partial Differential Equations in Physics. 1949 (Lectures
on Theoretical Physics, Volume V I ) 2: REINHOLD BAER.Linear Algebra and Projective Geometry. 1952 BUSEMANN A N D PAUL KELLY.Projective Geometry and Projective 3 : HERBERT Metrics. 1953 4 : STEFAN BERCMAN A N D M. SCHIFFER. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. 1953 5 : RALPHPHILIP BOAS,JR. Entire Functions. 1954 6: HERBERT BUSEMANN. The Geometry of Geodesics. 1955 7 : CLAUDE CHEVALLEY. Fundamental Concepts of Algebra. 1956 8: SZE-TSEN H u . Homotopy Theory. 1959 9 : A. M. OSTROWSKI. Solution of Equations and Systems of Equations. Second Edition. 1966 10: J. DIEUDONN~. Foundations of Modern Analysis. 1960 Curvature and Homology. 1962 11 : S. I. GOLDBERG. HELCASON. Differential Geometry and Symmetric Spaces. 1962 12 : SICURDUR 13 : T. H. HILDEBRANDT. Introduction to the Theory of Integration. 1963 14 : SHREERAM ABHYANKAR. Local Analytic Geometry. 1964 L. BISHOPANn RICHARDJ. CRITTENDEN. Geometry of Manifolds. 1964 15 : RICHARD 16: STEVENA. GAAL.Point Set Topology. 1964 17 : BARRY MITCHELL.Theory of Categwies. 1965 18: ANTHONYP. MORSE.A Theory of Sets. 1965 19: GUSTAVE CHOQUET. Topology. 1966 20: 2. I. BOREVICH A N D I. R. SHAFAREVICH. Number Theory. 1966 21 : Josh LUISMASSERA A N D J U A N JORCE SCHAFFER. Linear Differential Equations and Function Spaces. 1966 D. SCHAFER. An Introduction to Nonassociative Algebras. 1966 22 : RICHARD Introduction to the Theory of Algebraic Numbers and 23: MARTINEICHLER. Functions. 1966 24 : SHREERAM ABHYANKAR. Resolution of Singularities of Embedded Algebraic Surfaces, 1966
I n prejoration:
FRANCOIS TREVES. Topological Vector Spaces, Distributions, and Kernels. OYSTEINORE.The Four Color Problem. PETER D. LAXand RALPHS. PHILLIPS. Scattering Theory.
RESOLUTION OF SlNGULARlTlES OF EMBEDDED ALGEBRAIC SURFACES
Shreeram Shan kar Abhyan kar Division of Mathematical Sciences Purdue University Lajayette, Indiana
ACADEMIC PRESS
New York and London
1966
COPYRIGHT 0 1966, BY ACADEMIC PRESSINC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT W R m E N PERMISSION FROM THE PUBLISHERS.
ACADEMIC PRESS INC. 1 1 1 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l
LIBRARY OF CONGRESS CATALOG CARDNUMBER : 66-26258
PRINTED IN THE UNITED STATES OF AMERICA
Dedicated to Professor Oscar Zaris ki Without his blessings who can resolve the singularities ?
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Preface
Some twenty years ago there appeared, in the Annals of Mathematics, the marvelous memoir of Zariski entitled: Reduction of singularities of algebraic three-dimensional varieties. Not only was a daring and ingenious solution of a difficult problem given in it, but so much of the technique invented for the solution has proved to be of such significance for algebraic geometry in general! Hironaka’s brilliantly energetic recent solution of the general resolution problem for zero characteristic constitutes, indeed, a high tribute to Zariski’s memoir. At present I am able to pay only a modest tribute to Zariski’s memoir by giving a self-contained exposition of it. This then is the primary aim of the monograph. A secondary aim is to partially extend some of the results to nonzero characteristic. The algorithm needed for such an extension has already been published in four papers, and it will not be repeated here. This monograph contains the geometric part of the argument. However, we do include an alternative simple version of the algorithm for zero characteristic thereby making the monograph self-contained for that case. Finally, the matter is so arranged that about half of the monograph can be used as an introduction to certain foundational aspects of algebraic geometry. My thanks are due to Annette Wortman for an excellent job of typing the manuscript. T h e work on this monograph was partially supported by the National Science Foundation under NSF-GP4248-50-395 at Purdue University; I am grateful for this support.
S. S. A.
September, 1966 Purdue University
Vii
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Contents
vii
Preface
1
0 Introduction CHAPTER 1.
1
2 3 4 5
Local Theory
Terminologv and preliminaries Resolvers and principalizers Dominant character of a normal sequence Unramified local extensions Main results
7 45 61 108
148
CHAPTER 2. Global Theory
155 192 223 233
6 Terminology and preliminaries 7 Global resolvers 8 Global principalizers 9 Main results CHAPTER 3.
Some Cases of Three-Dimensional Birational Resolution
10 Uniformization of points of small multiplicity 1 I Three-dimensional birational resolution over a ground field of characteristic zero 12 Existence of projective models having only points of small multiplicity 13 Three-dimensional birational resolution over an algebraically closed ground field of characteristic # 2, 3, 5
238 26 I 262 283
Bibliography
283
Index of Notation
287 289
Index of Definitions ix
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$0. Introduction
Let k be a perfect ground field of characteristic p , and let X be a nonsingular irreducible three-dimensional projective algebraic variety over k. Then the principal results proved in this monograph are:
Global resolution. Given any algebraic surface Y over k embedded in X, there exists a sequence X + X , -+ X , + ... + X,, + X of monoidal transformations with nonsingular irreducible centers such that the total transform of Y in X' has only normal crossings and the proper transform of Y in X' is nonsingular. Global principalization. Given any ideal 1 on X, there exists a sequence X + XI + X , -+ ... + X , -+ X' of monoidal transformations with nonsingular irreducible centers such that the inverse image of I on X' is locally principal. Dominance. Given any irreducible projective algebraic variety X* over k such that X * is birationally equivalent to X, there exists a sequence X -+ XI + X , -+... + X , + X' of monoidal transformations with nonsingular irreducible centers such that X dominates X * . Birational invariance. If k is algebraically closed and X* is any nonsingular irreducible projective algebraic variety over k such that X * is birationally equivalent to X, then h i ( X ) = hi(X*) for 0, where h i ( X ) denotes the vector space dimension over k all i of the ith cohomology group of X with coefficients in the structure sheaf, and hence, in particular, the arithmetic genus of X = the arithmetic genus of X * . Uniformization. Assume that either p = 0, or k is algebraically closed and p # 2, 3, 5 . Let K be any three-dimensional algebraic function field over k and let W be any valuation ring of K containing k. Then there exists a projective model of Klk on which the center of W is at a simple point. Birational resolution. Assume that either p = 0, or k is algebraically closed and p # 2, 3, 5 . Let K be any three-dimensional 1
2
$0. INTRODUCTION
algebraic function field over projective model of KIA.
K. Then there exists
a nonsingular
History. The following version of global resolution was proposed by Levi [16] and proved by Zariski [25]: if p = 0 and Y is any irreducible algebraic surface over K embedded in X, then there exists a sequence X 4XI -+ X , -+ ... -+ X , -+ X' of monoidal transformations with nonsingular irreducible centers such that the proper transform of Y in X is nonsingular. For p = 0, Zariski [25] proved dominance. For p = 0, Zariski [23] proved uniformization for function fields of any dimension. For p = 0, Zariski [25] deduced birational resolution from uniformization and global resolution (in the form just mentioned). For p = 0, Hironaka [I51 generalized all the above six results to varieties of any dimension. What we have called global principalization corresponds to what Hironaka [15] has called trivialization of a coherent sheaf of ideals. We now describe the contents of the various chapters. Chapter One. In this chapter we prove a certain local version of global resolution which may be called resolution, and from it we deduce a certain local version of global principalization which may be called principalization; it may be noted that for this deduction it is necessary to have resolution without assuming Y to be irreducible. In 51 and 92, we establish the terminology and make some general observations concerning the basic concepts. In 53 we prove a theorem (see (3.21)) which corresponds to what Zariski [25] has called the dominant character of a normal sequence, and which says that if the multiplicity of a given point of the embedded surface Y can be decreased by monoidal transformations of a certain type then it can also be decreased by monoidal transformation of a more restricted type; this has the effect of reducing the proof of resolution to an apparently weaker assertion. In $4 the proof of this weaker assertion is further reduced (see (4.22)) to a certain statement (*) concerning monic polynomials in an indeterminate with coefficients in a two-dimensional regular local domain. The proof of resolution depends on the algorithm developed in the papers [5], [7], [8], and [9]; however, the matter is so arranged that this dependence is reduced to a
§O. INTRODUCTION
3
single point; namely, a part of [9: Theorem 1.11 is restated as (5.1) which is nothing but the said statement (*). In $5, the main results of the chapter are deduced as direct consequences of (5.1), (4.22), (3.21), and the preliminary considerations made in $1 and $2. For the purpose of comparison, in $5 we give an alternative simple proof of (5.1) for p = 0 which does not in any way depend on the papers [ 5 ] , [7], [8], and [9]; instead it uses the trick of killing the coefficient of Ze-l in a polynomial of degree e in an indeterminate 2; this trick was effectively used by Hironaka in [15], and it was also used by Abhyankar and Zariski in [lo]. As far as the case of p = 0 is concerned, the said alternative proof of (5.1) has the effect of making the entire monograph independent of the papers [5], [7], [8], and [9]. Chapter Two. $6 contains some generalities on the language of models. In $7 we show that resolution implies global resolution. In $8 we show that principalization implies global principalization, and that global principalization implies dominance. I n deducing birational resolution from uniformization and global resolution, Zariski [25] made use of the theorem of Bertini on the variable singularities of a linear system; in doing so he had to apply global resolution to a generic member of a linear system and hence to a surface not defined over k but defined over a pure transcendental extension k* of k . Now for p # 0 this approach causes two difficulties; namely, in the first place the said theorem of Bertini is then not valid and in the second place k* will not be perfect. However, in $8 we show that retaining a part of Zariski’s argument but replacing the use of Bertini’s theorem by the use of global principalization (as suggested by Hironaka) and without extending the ground field k, it is possible to deduce birational resolution from uniformization for any p . We refer to Serre [22] for the definition of the cohomology groups and for the result that the hi are finite and their alternating sum equals the arithmetic genus as classically defined in terms of the Hilbert polynomial; and we refer to Matsumura [17] for the result that: if k is algebraically closed and X and X * are any irreducible nonsingular projective algebraic varieties over k such that X and X* are birationally equivalent and X* dominates X , then h i ( X ) hi(X*) for all i >, 0, and if moreover X * is a monoidal transform of X with a nonsingular irreducible center then hi(X) = hi(X*) for all i 2 0;
birational resolution.
Chapter Three. As said above, for p = 0, Zariski [25] deduced birational resolution from uniformization and global resolution. I n 510 we show that, for p = 0,uniformization can also be deduced from resolution; it may be noted that for this deduction it is necessary to have resolution for the total transform and without assuming Y to be irreducible. I n $11 we state the resulting theorem: birational resolution for p = 0; thus, in view of the above-mentioned alternative proof of (5.1) for p = 0, we shall have completely reproved this theorem without using any results from the papers [5], [7], [8], and [9], and without appealing to Zariski’s paper [23] on uniformization. Actually what we show in $10 is somewhat stronger; namely, assuming resolution, we prove uniformization under the hypothesis that: the residue field of the given valuation ring W is algebraic over k and there exists a projective model of K/k on which the center of W is at a point of multiplicity e such that e! $ 0 modp. Consequently we would have birational resolution also for p # 0 if we could find a projective model of K/k such that every algebraic point of it has multiplicity < p . In 512 we show that it is possible to find such a model provided k is algebraically closed and p # 2, 3, 5. In 513 we state the resulting theorem: birational resolution when k is algebraically closed and p # 2, 3, 5. What we actually prove in 912 is this: assume that k is algebraically closed and let L be an algebraic function field over k of any dimension n; then there exists a projective model of L/k such that every rational point of it has multiplicity 1 where S = A,. Let S' be the valuation ring of ord, , and let R' E 2U(A, P ) such that R' dominates A . Then (1) S' E %(R') and S 4 B(R'). Moreover, ( 2 ) if P, is a nonzero prime ideal in A such that AIP, is regular and R' E 2U(A, P,) then P, = P.
PROOF. By ( I .4) we know that S' E %(R').Since dim S > 1 = dim S', we get that S # S'. Since S' E %(R')C W(A,P), W(A, P ) is an irredundant premodel of the quotient field of .4, and S' dominates S, we get that S 4 2U(A, P). This proves (1). To prove (2) let P, be a nonzero prime ideal in A such that A/P, is regular and R' E $W(A,P,). Let S , = A P 1. Since dim S > 1, we get that P is not a principal ideal in A and hence R' # A by (1.9.6) because R' E U ( A ,P) ; since R' # A and R' E W(A,P,), again by (1.9.6) we get that PI is not a principal ideal in A and hence dim S, > 1.
1. LOCAL THEORY
30
Now S' E %(R') C m(A,P,), W ( A , P,) is an irredundant premodel of the quotient field of A, S' # S , and S' dominates S ; hence S 4 m(A,P,). Since S 4 W ( A , P,) and S E B(A), by (1.9.5) we get that P,S # S and hence S C S,; by symmetry we get that S, C S. Therefore S, = S and hence P , = P. (1.10). Let R be a local domain, let S E %(R) with dim S > 0, let J be an ideal in R, and let V be a valuation ring of the quotient field of R dominating R. By a monoidal transform of ( R , S) we mean an element in IU(R, R n M(S)) dominating R. Since W(R, R n M ( S ) ) is a projective model of the quotient field of R over R, there exists a unique element R* in "(R, R n M ( S ) ) such that V dominates R*; clearly R* dominates R and hence R* is a monoidal transform of (R, S); R * is called the monoidal transform of (R, S) along V. Given a monoidal transform R' of (R, S), we define the (R, S, R')-transform of ] to be the ideal in R' generated by the set of all elements r in R' such that rxdE J for some nonnegative integer d and some element x in R' for which xR' = (R n M(S))R'. By a monoidal transform of (R, J, S) we mean a pair (R', J') where R' is a monoidal transform of (R, S) and J' is the (R, S, R')-transform of I. By the monoidal transform of (R, J , S) along V we mean the pair (R*, J*)where R* is the monoidal transform of (R, S) along V and ]* is the (R, S, R*)transform of J. By a quadratic transform of R we mean a monoidal transform of (R, R). By the quadratic transform of R along V we mean the monoidal transform of (R, R) along V. In this chapter we shall use monoidal transforms (which are not quadratic transforms) only when R is regular, S has a simple point at R, and J is a nonzero principal ideal in R; note that in this case the considerations of (1.4) apply. Given a regular local domain R, by an iterated monoidal transform of R we mean a local domain R* such that there exist finite such that: m is a nonnegative sequences (R,)oc,, 0. This being so for 1 < i n, we get that y E x-lA. Thus I-' = x-'A, and hence IF1= Ix-' and (IF1)$ = I.
prin,I C y A I C y A 3 y-l E I-' 3 Iy-' C ZIP1. Assuming that prin,I C y A , we shall show that (Iy-l)(Zy-l)-l = II-' and this will complete the proof. We can take x E A such that xA = prin,I. Then xA C y A and hence x / y E A ; since I C xA, we get that Iy-' C (x/y)A. For any nonzero element z in A we have that: Iy-' C z A + I C zyA + prin,IC zyA * xA C zyA + (x/y)AC zA. Therefore prin,(Iy-l) = (x/y)A, and hence by (1.1 1.7) we get that (Iy-l)(Iy-l)-l = (Iy-')(x/y)-l = Ix-1 = 11-1. 3
(1.12). By a semiresolver we mean a sequence (R, , J , , Si)O.c,i<er where: either m is a positive integer or m = m; R, is a regular local domain, Ji is a nonzero principal ideal in R , , and Si is a positive-dimensional element in E ( R i , J,) having a simple point at R, for 0 i < m ; ( R , , J,) is a monoidal transform of (Ri_l , Sip')for 0 < i < m ; and for 0 i < m we have that: dim S, = 2 e E2(Ri,1,) has a strict normal crossing at R , and E2(Ri 9 J i ) # B . By an infinite semiresolver we mean a semiresolver ( R , , J , , SJ0 ,,, where m = 00 and ( R , , Ji) is unresolved for 0 i < co. By an finite semiresolver we mean a system [(R, , J , , S.) z O-,i<m 9 (Rn?, J J ] where: m is a positive integer; ( R , , J , , S i ) o ~ i < nisha i < m ; R , is s-miresolver such that (Ri , Ji) is unresolved for 0 a regular local domain and Jn, is a nonzero principal ideal in R , such that (R?,&, J1,J is resolved; and (R,,,, J,,,) is a monoidal transform of (Rrn-1 9 Jm-1 9 Snt-1). By a finite weak semiresolver we mean a system [ ( R , , J , , Si)o 1 , 1 , (R,,,, J J ] where: m is a positive integer; Ri is a regular local domain and Ji is a nonzero principal ideal in Ri for 0 i m ; Siis a positive-dimensional element in E(Ri, J,) having a simple point at Ri for 0 i < m ; ( R , , Ji) is a monoidal transform of ( R i - l , Ji-', Sip') for 0 < i m ; and for 0 i < m we have that: dim Si= 2 E2(Ri, J,) has a strict normal crossing at R, . By a resolver we mean a sequence ( R i , Ji ,I, , Si)O.,icn,where: either m is a positive integer or m = CO; Ri is a regular local
0 for u < i 3, and hence xi E R, n M ( S ) for u < i 3. Since dim S = 2, we get that u 2 1, and if u = 1 then R, n M ( S ) = (xZ,~ 3 ) R o .Also, if u = 2 then R , n M ( S ) = ( y * , ~ 3 ) R ofor some y * E R, (namely, since ordR,x3 = 1 and x3 E R, n M ( S ) = (y', x')R, , we get that xg = YO' r2z' where rl and r2 are elements in R, at least one of which is not in M(R,); take y* = z' in case Y , 4 M(R,), and y* = y' in case rl E M(R,)). Let w = xyl ... x2. Then w E R , and w 4 R , n M ( S ) . Let
<
+
+
>
(3.3). Let R, be an n-dimensional regular local domain and let (Ri)o j we have that E' = E n B(Ri) and E' has a normal crossing at Ri . (3.3.2). Assume that R, is pseudogeometric. Then E' contains at most one element and there exists a nonnegative integer j such that for all i 3 j we have that E' = E n B(R,) and E' has a normal crossing at Ri .
PROOFOF (3.3.1). Clearly E n %(Rb)C E n %(R,) whenever b a, and hence there exists a nonnegative integer c such that E' = E n %(Ri) for all i 3 c. Consequently if E' = PI then we have nothing to show. So now assume that E' # 0 and take S EE'. Then there exists a basis ( x l , ..., x,) of M(R,) such that R, n M ( S ) = (xZ, ..., x,)R, . Repeatedly applying (1.10.10) we get that for all i 0 we have that dim Ri = n, M(R,) = ( x l , x,/xl , ..., x,/xi)Ri, and R, n M ( S ) = (x,/x:, ..., xn/xi)Ri. I n 0. It now suffices particular S has a simple point at R, for all i
>
>
1. LOCAL THEORY
64
to show that E’ = {S}. Suppose if possible that E’ # {S}and take S‘ E E‘ with S‘ # S. For any i > 0 and any z E Ri-l n M ( S ’ ) , by (1.10.9) we have that z/xl E Ri n M ( S ’ ) . Therefore for any z E R, n M ( S ’ ) we get that z/x: E R, n M(S’) for all i 2 0 and hence z/x: E M ( R i ) for all i 0. Since S‘ # S, there exists z E R, n M ( S ’ ) such that z 4 ( x Z , ..., xn)R, . Let h: R, + R,/(x, , ..., xn)R, be the canonical epimorphism and let i = ord,,(,o,h(z). rzxz r,x, Then i is a nonnegative integer and z = rlx: where r l , ...,r,, are elements in R, such that rl $ M(R,). Now z/xf = rl r2(xZ/xf) * * . r,(x,/xf), rq E Ri for 1 q n, xq/x, E M ( R i ) for 2 ,< q ,< n, and r1 4 M ( R i ) . Therefore z/x: $ M(R,) which is a contradiction.
>
+ +
+
+
+ +
<
, c. Consequently if E’ = 0 then we have nothing to show. So now assume that E’ # 0 and take S & E’. If dim R, # n for some d then Ri is an ( n - 1)-dimensional regular local domain for all i d, and hence E’ = {S}and E‘ has a normal crossing at R, for all i d. So also assume that c such that dim Ri = n for all i 2 0. By (3.2) there exists e every element in E’ has a simple point at R e . Now by (3.3.1) we get that E’ contains at most one element and there existsj 2 e such that E’ has a normal crossing at R, for all i j.
b E’
> a, and hence there exists a nonnegative >
>
>
>
(3.4). Let R, be an n-dimensional regular local domain and let J, and I, be nonzero principal ideals in R, such that I , has a quasinormal crossing at R, .Let (Ri , Ji , be an infinite sequence such that for 0 < i < GO: Ri is a regular local domain, Ji and Ii are nonzero principal ideals in R, , and (R, , Ji ,I,) is a monoidal transform of (Ri-l , JiPl, IiPl , RiWl).Let E be a finite set of ( n - 1)-dimensional m
elements in %(R,), and let E’ following.
=
E n(
n %(Ri)).Then we have the i=O
(3.4.1). Assume that every element in E has a simple point at R, . Then E’ contains at most one element and there exists a nonnegative integer j such that for all i >, j we have that E = E n %(R,)and ( E ,I,) has a pseudonormal crossing at R, .
53. DOMINANT CHARACTER
65
OF A NORMAL SEQUENCE
(3.4.2). Assume that R, is pseudogeometric. Then E' contains at most one element and there exists a nonnegative integer j such that for all i 2 j we have that E' = E n B(Ri) and (El, Ii) has a pseudonormal crossing at R, .
PROOF OF (3.4.1). By (1.10.8) we know that I, has a quasinormal crossing at Rifor all i >, 0. By (3.3.1) we know that E' contains at most one element and there exists a nonnegative integer j ' such that E' = E n B(R,) for all i 2 j ' . If E' = 0 then it suffices to take j = j ' . So now assume that E' # 0 and take S E E'. Then E' = {S}and there exists a basis (xl, ..., x,) of M(R,) such that R, n M ( S ) = ( x Z , ..., xn)R, . Repeatedly applying (1.10.10) we get that for all i 2 0 we have that dim R, = n, M(R,) = (xl , x2/xi , ..., x , / x f ) R , , and R, n M ( S ) = (x2/xf,..., x , / x i ) R , . Now I, = z1... z,Ro where z1, ..., z,, are elements in R, with q a (we take z1... z,Ro = R, in case ordR,zq = 1 for 1 a = 0). Upon relabeling zl,..., z, we may assume that Z ~ (Ex 2 , ..., xn)R, for 1 q b, and z q $ ( x 2 , ..., xn)R, for b , 0. Clearly Ii = zIi for all i 0, and hence (S,I d ) has a pseudonormal crossing at R, for all i e. I t now suffices to take j = max(j', e).
< < <
1 and assume that the assertion is true for all values of n smaller than the given one. If m* = 0 then a(i) = i for 0 i < n, and it suffices to take (Ri , Ji) = (Ri , Ji) for 0 i < n. So now assume that m* # 0. Let c(O), c(l), ..., c(n - 2) be the permutation of 0, 1, ..., n - 2 defined thus: if n - 1 4 W then c(i) = a(z) for 0 < i < n - 2; and if n - 1 E W then c(i) = u(z3 for O , < i , < m - 2 , and c ( i ) = a ( i + I ) for m - - 1 G i G n - 2 .
1 and hence ordR.PIR' > 1; since S' E (E(R',PIR'), we deduce that S' E G(R', P,R'); this contradicts (1). Therefore Pl # R n M(S' ); since Pl C R n M(S'), we get that dim RRn,(s,, 2 2 and hence by (4.2.1) we get that dim RRnM(s#)= 2. Therefore in view of (I), by (4.2.3) we get that ( R n M(S'))S' = M(S') and hence M(RRn,(s*))S' = M(S'). Consequently RRnMot)E E2(R, J).
< < <
1 in an indeterminate 2 with coeficients in S. Let r E S and let s = tx;l ... x 2 where t is a unit in S and a , , ..., a, are nonnegatiwe integers.L e t g ( 2 ) = sr"f(s2 r). Assume t h a t g ( 2 )E S[ZJ and 0 < ord,g(Z) < e. Then we hawe the following. (4.10).
(x,
, ..., x,)
+
(4.10.1). Let r' E S and let s' = t'x! where 0 # t' E S and b is a nonnegatiwe integer. Assume that s'-"f(s'Z r') E S[Z]. Then
b ,< a, and (r'
+
- r)/xi E
S.
(4.10.2). Let Y' E S a n d let s' = t'x!1 ... xin where t' is a unit in S and b, , ..., b, are nonnegatiwe integers. Let g ' ( 2 ) = s'-ef(s'Z r'),
+
1. LOCALTHEORY
1 I8
...
t* = (t’ - Y ) / s ’ , ~i = a, - b, , t* = t/t’, and s* = t*xil x?. Note that then c1 , c, are integers, t* is a unit in S , and g ( 2 ) = s*-“g’(s*Z - t * ) . Assume that g’(2) E S[Z]. Then t * E S and Ci>0fOt 1 < i < n .
...,
PROOF, (4.10.2) follows from (4.10.1). We shall now prove (4.10.1). Let G(2) = t e g ( F 2 ) . Then (1)
G(2)
Also ord,G(Z)
= Ze =
+ GIZe-l + ... + G,
G, E S.
ord,g(Z) and hence G, E M ( S )
(2)
and there exists an integer d with 1 (3)
with
ordsGd
1, and u = 1 if n = 1. Let f’(2)= s’-ef(s’Z + t‘) and f*(2)= t’”f’(t‘-’Z). Then f*(2)E S[Zl and
+
f”(2)= (uxY-~)‘G(u-’.x-~Z
(4)
Let R
=
(I’
- Y)U-*X;~).
SS,,.By (1) and (4) we get that f*(O)4*
=
G* where
e
G*
= (I’
- I)”
+ C G~$X?(Y’
- Y)+~;
i=1
since f*(z) E s[q, we get that ordRf*(0)@ 2 eb; also, if ord,(r’ - I ) < a then clearly ord,(r’ - t y = ord,G*; therefore we get that (5)
if
OrdR(Y’ - I)
a. Then by ( 5 ) we get that ord,(r‘ - I ) 2 a, i.e., (t’- t ) / x : E S. Let h: S + S/xlS be the canonical epimorphism. Then there exist elements I * and t” in S such that (t’- t ) / x y = I * t”xl and ord,r* = ordh(,,h((t’ - t ) / x y ) ; note that then ord,r* = ordhcs,h(r*). Let F’(2)= x 1( ~ - ~ - ’.f) ”*(x,l+n-bZ) and F ( 2 ) = F’(Z I ” ) .
a,
F ( Z ) = Z e + FlZe-I
(6)
119
EXTENSIONS
we get that F'(Z) E S [ Z ] and hence
+ .*.+ F,
with FiE S.
By (4)and the definition of F'(2) and F ( 2 ) we get that
+
F ( Z ) = ~ ~ x ; ~ G ( u - ' x , Z Y*u-').
(7)
By (I), ( 6 ) , and (7) we get that c
Geue = G(0)ue
= x;F(-y*x;l)
= (-l)er*e
+C (-l)e-i~xi~*e-i i l
i=l
and hence ordh(s)h(r*e) = ord,(,&G,ue); consequently in view of (2) we get that ord,(,)h(r*) > ordh(Jz(u); now ord,(,)h(r*) = ord,r* and ordh(s)h(u) 2 ord,u; therefore (8)
For 0
ordsr*
< i < j < e let
> ordsu.
Wtj be the elements in S such that
Then by (I), (6), and (7) we get that Fdx;' = G,ud
(9)
(1 1)
+C
Wod~*d
WidGi~i~*d-i.
i=l
Let p = ord,F, and q get (10) and (1 1): (10)
d-1
+
q
for j
>0
+
+
+
we get that si E S* for all j 2 0 and s = so slz s2z2 * * * ; since s E S C S*, by the uniqueness part of (1) we get that s = so; clearly so E M(S*)b and hence s E S n M(S*)* = M(S)b; thus ( z R M(R)b)n S C M(S)*.This completes the proof of (4.1 1.1). (4.1 1.2) follows from (4.1 1.1). T o prove (4.1 1.3) let f ( Z )be any polynomial in an indeterminate Z with coefficients in the quotient field of S such that f(2) E R; then f(2)= (fo/t)+ (f,/t)Z (fe/t)Zewhere e is a nonnegative integer, t is a nonzero element in S, and fo , ...,fe are elements in S ; since f(z)E R, by (1) there exist elements g o ,g, ,g, , ... in S* such that f(z)= go g,z g2z2 * - * ; now fo f1z . * * +fez" = t f ( z ) = (tgo) (tgl)z (tg2)zz and hence by the uniqueness part of (1) we get that fi = tgi for 0 j e; thus f , E ( t S * ) n S = tS for 0 j e and hence h/tE S for 0 j e, i.e., f(2)E S[q.
+
+ +
+ +
<
0. Therefore z / x E R'. Consequently J'L' = ( w / e ) R ' and I' = xp+eyQL'.Let f be a coefficient set for R. Suppose if possible that c* = c ; then ordR,J'L' = e = Ord,]L; for 1 < j e let r j be the unique element in f such that g j - r j E M(R); since g, E M(R), we get that re = 0 and hence
+
+ +
0 and hence in particular J' # R'. Let L' be the (R, S, R')-transform of L. Then J'L' is the (R, S, R')-transform of JL. Suppose if possible that xIz E R'; then J'L' = (w/z")R'; now
1; consequently J'L' = R' and hence J' = R'; this is a contradiction. Therefore z/x E M(R'). Consequently dim R' = 3, R' is residually rational over R, M(R') = (x, y, z/x)R', J'L' = (w/x")R', and I' = xP+~~*L'. Now e
w/xe = (z/x)L
+1
gjx'"-l)jybj(z/x)"-i
.
i=l
Let h': R' + R'/(z/x)R'. Then by (4.17) we get that ord,,gj = ord,,,,,,h'(g,) for 1 < j < e, and ord,,gi, = ord,gi, < j ' . Also g, E M(R'). Since (u - 1) + b < a + b, by the induction hypothesis we conclude that one of the following two conditions is satisfied.
, (1') There exists a finite weak resolver [(R, , J, ,I, , Si)oGi..v (R, , J, ,I,)] such that: (R, , J, ,I,) = (RT, JT, I T); dim S, = 2 for 0 a' < a ; dim R, = 3 and ord,, Ji = d > ordR,Jq, for 0 a' < a ; and V dominates R, .
. Then g*(Z) E S * [ Z ] and ord,*g*(Z) = ordpg(Z). Consequently g*(Z) = 2” +glZ“-’ ... +g,
+
+
+
+
+
+
+
where g, , ...,g, are elements in S* such that g, E M(S*), and gi, 4 M(S*)j’ for some j ’ with 1 j’ e. I n particular then g, , ...,g, are elements in R* with g, E M(R*), and in view of (4.1 1.2) we get that ordR*gj= ordhcR*,gj for 1 j e where h: R* --t R*Iz*R* is the canonical epimorphism, and ord,pgj, < I r .
<
ordRmJnlfor 0 < i < m, and V dominates R,n . This however follows from (4.21.5). (4.23). REMARK.If in (4.22) we only wanted to prove the weaker assertion that R is weakly semiresolvable then, upon disregarding several considerations of this section and simplifying some of the remaining considerations, we can make a simpler proof. The reader may find it instructive to extract such a simpler proof of the said weaker assertion.
$5, Main results I n [!I Theorem : 1.11 we proved the following.
(5.1). Let So be a two-dimensional regular local domain such that So/M(So) is an algebraically closedfield having the same characteristic as S o . Let ( x o ,yo) be a basis of M(So), let t be a coeficient set for So,and let f ( 2 ) be a monic polynomial of degree e > 0 in a n indeterminate 2 with coeficients in So. Let ( S , , xi ,y,)o, 2 (note that by (6.4.6) and (6.4.7) we know that 3(II-l) is a closed subset of X ; also note that if X is nonsingular then X is normal).
172
2. GLOBAL THEORY
This follows from (6.4.8) by noting that every one-dimensional normal local domain is a principal ideal domain (see [27: 46 and 47 of Chapter V ] ) . (6.4.10). Let I be any nonzero principal ideal on any model X of K/k. Then 3 ( I ) is either empty or pure 1-codimensional (note that by (6.4.6) we know that 3 ( I ) is a closed subset of X).
This follows from Krull’s principal ideal theorem [27: Theorem 29 on page 2381. (6.4.11). Let X and X’ be any models of K/k such that X’ dominates X , and let I be any ideal on X . Then I X ‘ is an ideal on X‘. Moreover, I = 0, o IX’ = O x , .
PROOF. Given any R’ E X ‘ let R = [X’, X ] ( R ’ ) .We can take an affine ring A over k such that R E %(A)CX.By (6.2.5) and (6.2.12) there exists an affine ring A’ over k such that R’ E %(A’) C [X’, X]-l(%(A)). Now A C [X’, X ] ( S )C S for all S E [X’, XI-l (%(A)), and hence in particular A C S for all S E %(A’). Therefore by (1.1 1.5) we get A C A’. Consequently (A A I)A’ is an ideal in A’, and clearly ((A n I ) A ’ ) S = I’S for all S E %(A’). Therefore by (1.11.5) we get that (A n I)A’ = A’ n 1’, and (A’ n I’)S = I’S for all S E%(A’).This shows that IX‘ is an ideal on X . By (6.4.7) we get that I = 0, o I X = O x , . (6.4.12). Let X and X‘ be any models of K/k such that X dominates X , and let I be any nonzero ideal on X such that IX‘ is principal. Then [X’, X]-’(3(1))is a closed subset of X’, and [ X ,X ] - l ( 3 ( I ) ) is either empty or pure 1-codimensional.
This follows from (6.4.2), (6.4.10), and (6.4.1 1). (6.4.13). Let X and X’ be any models of K/k such that X’ dominates X , and let Z be any closed subset of X . Then 3([X’, X]-l(Z), X’) = rad(3(Z, X ) X ’ ) .
PROOF. By (6.4.6) we get that 3(2,X) is an ideal on X, and 3(3(Z,X)) = 2. Hence by (6.4.2) we get that [X, X ] - l ( Z ) = 3(3(Z,X)X’), and by (6.4.11) we get that 3(2,X ) X ’ is an ideal
$6. TERMINOLOGY AND PRELIMINARIES
173
on X'. Therefore by (6.4.6) we get that 3 ( [ X X]-l(Z), , X') = rad(3(2, X ) X ' ) . (6.4.14). Let X and X' be any models of K / k such that X' dominates X , let Y be any subset of X , let Z be any closed subset of X , let I be any nonzero ideal on X , let Y' = [ X ,X]-'(Y), let 2' = [ X ' , X]-l(Z), and let I' = I X . Assume that every irreducible component of 3(II-l) having a nonempty intersection with Z n Y is contained in 2. Then every irreducible component of 3(I'I'-l) having a nonempty intersection with 2' n Y' is contained in Z'. (Note that by (6.4.6), (6.4.7), and (6.4.11) we know that 3(IIP1) is a closed subset of X , and 3(I'P1)is a closed subset of A?.)
PROOF. Let Z* be any irreducible component of 3(I'I'-l) having a nonempty intersection with 2 n Y'. Take R' E Z* n 2' n Y' and let R = [X,XI(R'). Since R' E 3(I'II-l), by (6.4.2) we get that R E 3(II-l). Since R' E 2 n Y', we also get that R E 2 n Y. Let S* be the generic point of Z*. Now S* E B(R'), and R' dominates R; therefore upon letting S = RRnM(S*)we get that S E %(R) C X , and S* dominates S ; consequently [X, X ] ( S * )= S ; since S* E 3(I'P1),by (6.4.2) we get that S E 3(II-l). Let 2, be an irreducible component of 3(IIP1)passing through S. Now 2, is a closed subset of X, and hence R E 2, ; therefore 2, n 2 n Y = 0 , and hence by assumption we get that 2, C 2; consequently S E 2, and hence S* E 2 . By (6.2.5) we know that 2' is a closed subset of X , and hence Z* C 2 . (6.5). For any ideal J on any model X of K/k we define: G ( J ) = (singular locus of J ) = {R E 3 ( J ) : R/(JR)is not regular}. In view of (6.4.4) and (6.4.6) we get the following. (6.5.1). For any model X of K/k we have the following: (1) If Z is any closed subset of X then G ( Z ) = G(3(2, X)). (2) If J is any ideal on X then G ( J ) = G(rad J) u { R E X :J R # rad(JR)}. (3) If J is any ideal on X and A is any afine ring over k with B(A) C X t h e n 6 ( J )n %(A)= G(A,A n J). Let J and I be any nonzero principal ideals on any nonsingular model X of K/k. Given any R E X and E C B(R), we say that (E, I )
174
2. GLOBALTHEORY
has a normal crossing at R if (E, IR) has a normal crossing at R. Given any R E X and E C B(R), we say that (E, I ) has a strict normal crossing at R if (E, IR) has a strict normal crossing at R. We say that I has only normal crossings if for each R E X we have that IR has a normal crossing at R. We say that (1,I ) has only quasinormal crossings if for each R E X we have that (JR, IR) has a quasinormal crossing at R. We say that I has only quasinormal crossings if for each R E X we have that IR has a quasinormal crossing at R; note that this is equivalent to saying that (I, lX) has only quasinormal crossings. Given R E X and S E B(R), we say that (S,I ) has a pseudonormal crossing at R if (S,IR) has a pseudonormal crossing at R. Given R E X and E C B(R), we say that (E, I ) has a pseudonormal crossing at R if (E, IR) has a pseudonormal crossing at R. We deJne: G*(J = { R E X : (R, JR) is unresolved}. J is said to be resolved if G*(J = 0. In view of (6.4.4) and (6.4.6) we get the following. (6.52). For any nonzero principal ideals J and I on any nonsingular model X of K / k we have the following: (1) G*(J) = G*(rad J = G(rad J) = G ( 3 ( J ) . (2) I has only normal crossings o rad I has only normal crossings o 3(I) has only normal crossings. ( 3 ) I has only quasinormal crossings o rad I has only quasinormal crossings o each irreducible component of 3 ( I ) is nonsingular. (4)If I has only normal crossings then I has only quasinormal crossings. ( 5 ) If (1,I ) has only ,quasinormal crossings and G*(J = 0 then ]I has only normal crossigs. (6) If I has only quasinormal crossings and G*(J = 0 then ]I has only quasinormal crossings.
In view of (6.2.10), (6.4.6),'(6.5.1),and part (1) of (6.5.2)we get the following. ! (6.5.3). Assume that for every a8ne ring A over k with quotient field K and every ideal Q in A we have that G(A, Q ) is closed in B(A) (see (1.2.6)). Let X be any model of K/k. Then we have the following: (1) If 2 is any closed subset X then G ( 2 ) is closed in X . (2)If J is any ideal on X then G ( J ) is closed in X . (3) If X is nonsingular and J is any nonzero principal ideal on X then G*(J is closed in X and codim G*(J 2 2.
96. TERMINOLOGY AND PRELIMINARIES
175
For any ideal J on any model X of K / k and any Z C X we define:
and: a*(Z, J) = {R E Z: ord,JR = ord,fl; note that: (1) if 2 # 0 then ord,J is either a nonnegative integer or co; (2) if Z # 0 and J = Ox then ord,J = CQ and a*(& J) = 2, (3) if 2 f 0 and J # Ox then: E*(Z, J) # 0 e-ord,] is a nonnegative integer. For any regular point R of any model X of K / k and any ideal J on X we deJne: a(R, J) = @(R,JR) and @(R, J) = @(R, JR) for every nonnegative integer i ; note that by (1.3.1) we then have that a(R, J) = (E*(B(R), J). From (0.4) we now deduce the following. (6.5,4), Assume that for every afine ring A over K with quotient field K and every ideal Q in A we have that G ( A , Q ) is closed in % ( A )(see (1.2.6)). Let J be any ideal on any model X of K/k, let Y be any open subset of X with G ( X )n Y = O , and let 2 be any nonempty closed subset of Y . Then E*(Z, J) is a nonempty closed subset of Z .
PROOF. Let W be the set of all nonempty closed subsets Z* of 2 such that (E*(Z*, J) = (E*(Z,J). Then W # o and hence by (6.2.15), W contains a minimal element 2'. Let Z , , ..., 2, be the irreducible components of 2 ,and let Si be the generic point of Z, . In view of (6.2.12) and (6.5.3) we can find an affine ring A , over K such that S, E B(A,) C Y and %(A,)n 6(Zi) = O . Upon taking ( A i , A , n J, A , n M(S,)) for ( A , J, Q) in (0.4), we can find an
ideal Hi in A , with Hisi = S, such that ord,JR = ord,giJSi for all R E %(A,)for which R C Siand HiR = R; upon letting Zi = { R E %(A,): R C S, and H,R # R} U (2, - %(A,)), in view of (6.2.9) and (6.2.12) we get that 2; is a closed subset of -2, with S, 6 2; , and clearly ord, JR = ordSiJSi for all R E Zi - 2;. Upon relabeling 2, , ..., 2, we may assume that ordzlJ = = ordzmJ > ordZtJ for m < i n; clearly 2, u ... U Z,* is a nonempty closed subset of Z and a*(Z, u ... u 2 , , J) = E*(Z', J); since 2' is a minimal element of W, we must have Z , u u Z , = Z and hence m = n. Thus ord,, J = ord,, J for
l
= Ad[(Ydb/Ydj)14bBn(d,1
9
= Ad[((XdaYdh)/(Xdiydj))l
G di j
< a < n r ( d ) .1< b < n ( d d .
Then by (6.6.5) we get that:
u m(&, u e
m(x, 1) =
A d
1) =
n(d)
e
I ) = d = l m(Ad
9
%(Edi);
i=l
d=l
mn(x,
u u
m(d)
m(&, A d
I),
A d PI
1)s
2u(Ad
uuu
1)= j = l
Ad
9
%(Fdj);
m(d) n(d)
e
m(x,zl) = d=l
i=l
%(Gdij)*
+l
Clearly Gdijis the smallest subring of K which contains Edi and Fdj . Therefore by (6.2.7) and (6.2.12) we get that m(A, , Ad n I) and m(A,, A, n J) are models of K/k for 1 d e, and
<
, n( V). For each V E %(Xo)we clearly have that V E [%(Xo),X,(,,)]-l(D(V)), and by (6.2.5) we get that [%(X,),X,(,,)]-l(D(V ) ) is an open subset of %(X,); now %(X,) is quasicompact by (6.2.13), and hence there exists a finite number of elements V , , ..., V , in %(X,,)such that
47. GLOBAL RESOLVERS
203
Let m be any nonnegative integer such that m q. Then clearly Ti = o for all i m.
> n( ifi) for I <j
(7.12). If K / k is locally strongly detachable then K/ k is globally detachable and globally strongly detachable.
PROOF. Follows from (7.10) and (7.11.2). (7.13). DEFINITION. By aglobalsubresolver of K/k we mean a sequence(&, J i , I i , Z i , Ti)Ogi<mwhere:(1)eithermisapositiveinteger or m = 00; (2) for 0 i < m: X i is a nonsingular model of K/ k,Ji and Ii are nonzero principal ideals on X i such that Ii has only quasinormal crossings, Ziis a pure 2-codimensional closed subset of X , with Zi C E*(G*( Ji), Ji),and Ti is a nonsingular closed subset of 2, such that for every R E T i , upon letting S be the generic point of the irreducible component of Ti passing through R, we have that (S,I i ) has a pseudonormal crossing at R and: dim S = 2 o E2(R,J i ) has a strict normal crossing at R and (2,n CC2(R,Ji), I i ) has a pseudonormal crossing at R ; and (3) for 0 < i < m: ( X i , Ji , I i ) is the monoidal transform of (Xi-l, I,-l) with center Ti, ordGq,,)Ji = ordG*(,,-,)J i - l , and 2, is the closure in Xi of { S E E*(G*( Ji), Ji): S dominates a two-dimensional point of ZiP1}. By an infinite global subresolver of K/k we mean a global subresolver (Xi, J i , I i , Zi,Ti)osi<wbof K/ k where m = 00 and Ti # 0 for infinitely many distinct values of i. By a jinite global subresolver of K/k we mean a system [(Xi, Ji , I t , Zi, Ti)o--i<m,( X ’ , J’, l’)]where: m is a positive integer; ( X i , Ji , Ii , Zi, Ti)Oii<m is a global subresolver of K/k such that Ti is irreducible for 0 i < m ; X is a nonsingular model of K/k and J’ and I ’ are nonzero principal ideals on X such that I’ has only quasinormal crossings; ( X ’ , J’, 1’) is the monoidal transform of (Xm-l, J l a _ l , Im-l) with center TmPl; and for every S E s*(J’) such that S dominates a two-dimensional point of Zm-l . we have that ord,J’S < ordGqJ,-,) K /k is said to be globally subresolvable if: given any nonsingular model X of K/k, any nonzero principal ideals J and I on X such that I has only quasinormal crossings, and any pure 2-codimensional closed subset Z of X with 2 C %*(G*( J ) , I), there exists a finite
229
<j
*
Now f(I) is a nonzero ideal in D, and hence by (12.1.8) there exists a nonzero ideal J' in E such that for every ideal Q' in E, upon letting v be the number of minimal prime ideals of Q' in E which do not contain J' and u be the number of minimal prime ideals of Q'D in D which do not containf(I), we have that u v. Let J* = g'(J'). Then J* is a nonzero homogeneous ideal in B . Take J = xJ*. Then J is a nonzero homogeneous ideal in B. Given any homogeneous ideal Q in B, let v = u( J , Q ) and let Q 1 , ..., Qv be the minimal prime ideals of Q in B such that Qi # B I B and J $ Q i for 1 i v . Since J $ Q i we get that x $Qi for 1 i v. It follows that g(Ql), ...,g(QJ are distinct minimal prime ideals of g(Q) in E, and J' $g(Qi) for 1 i v. Therefore there exist distinct minimal prime ideals Pi,..., PL of g(Q)D in D such that f(1)$ Pi for 1 \< i v . Let Pi= f'(Pi) for 1 i v . Then P, , ..., P, are distinct minimal prime ideals of Q A in A , and x $ Pi and I $ Pi for 1 i v. Since x 4 Pi we get that Pi # A,A for 1 i v. It follows that u(1, Q A ) >, v , i.e., ~ ( 1Q,A ) 2 u( J , Q).
<
...,fm)A = Pl n ... n P,, ; C
+ (rad(flA))
+ (rad(f,A))
274
3. THREE-DIMENSIONAL BIRATIONAL RESOLUTION
consequently w v(i)
0
P& c Pl n
n P,,.
i=l j=1
Also W
(.A , ...,MA
= n (p: 4is1
and hence
Thus p1n
n P, =
P& (-1 j=1
and hence for each q with 1 some (i,j ) . Therefore
< q < u we have that Pp = P& for
(12.3.2). Let d be any positive integer and let L be any A,-subspace of Aid). Then LA(d))< Q(A)& where q = [L : A,].
PROOF.By (12.3.1) we get that u(A,LA) , u. Let = {y, Y,-,z y,-dw: y i € Bi for n - w i n}, and let EA-" = (0). T h e n En[-11 C E!] C Ebl C ... C = A, are A,-subspaces of A,, and hence
+ +
+
EL~] < < Ekl
c [Eplpp-'] A,]. n
[An : A,]
=
:
w=o
<
1 and assume that the assertion is true for all values of r(A) smaller than the given one. Since A, is algebraically closed, by the Hilbert Nullstellensatz [28: Lemma on page 1651 there exists R E m(A) such that R is residually rational over A,. Now by (12.1.6) there exists a homogeneous subdomain B of A such that r(B) = r(A) - 1, B,A is a nonmaximal homogeneous prime ideal in A, and %(A, B,A) = R. Since rad(B,A) # A,A, by (12.3.4) we get that g(A) t(A) 2 g(B) t(B) 1. By the induction hypothesis we have that g(B) t(B) >, r(B). It follows that g(A) t(A) 2 ;(A).
+
+ +
+
+
(12.3.6). Let t = t(A) and g = g(A). Assume that A, is algebraically closed, and let I be any n w z e r o homogeneous ideal in A. Then there exist A,-subspaces L, L, , ..., L, of A, such that: L C L, n ..-nL, ; [L : A,] = t ; {L,A, ...,L,A} is the set of all minimal prime ideals of LA in A; L,A, ..., L,A are distinct; L A = (L,A) n
278
3. THREE-DIMENSIONAL BIRATIONAL RESOLUTION
... n (L,A) n T where either T = A or T is a homogeneous ideal in A which is primary for A,A; and for 1 i < g we have that [L, : A,] = r(A) - 1, I q! L,A # A,A, L,A is not contained in any homogeneous prime ideal in A other than L,A and A,A, %(A,L,A) is a t-dimensional regular local domain which is residually rational over A, , and if (s, , ..., s f ) is any A,-basis of L and si is any element in A, - L, then (s,/s; , ..., s,/s$I(A, &A) = M(%(A,L,A)).