Algebraic Geometry TATA INSTITUTE OF FUNDAMENTAL RESEARCH STUDIES IN MATHEMATICS General Editor : M. S. Narasimhan 1. M. Herve´ : S everal C omplex V ariables 2. M. F. Atiyah and others : D ifferential A nalysis 3. B. Malgrange : I deals of D ifferentiable F unctions 4. S. S. Abhyankar and others : A l g e b r a i c G eometry ALGEBRAIC GEOMETRY Papers presented at the Bombay Colloquium 1968, by ABHYANKAR ARTIN BIRCH BOREL CASSELS DWORK GRIFFITHS GROTHENDIECK HIRONAKA HIRZEBRUCH IGUSA JANICH¨ MANIN MATSUSAKA MUMFORD NAGATA NARASIMHAN RAMANAN SESHADRI SPRINGER TITS VERDIER WEIL Published for the tata institute of fundamental research, bombay OXFORD UNIVERSITY PRESS 1969 Oxford University Press, Ely House, London W. 1 glasgow new york toronto melbourne wellington cape town salisbury ibadan nairobi lusaka addis ababa bombay calcutta madras karachi lahore dacca kuala lumpur singapore hong kong tokyo Oxford House, Apollo Bunder, Bombay 1 BR © Tata Institute of Fundamental Research, 1969 printed in india INTERNATIONAL COLLOQUIUM ON ALGEBRAIC GEOMETRY Bombay, 16-23 January 1968 REPORT An International Colloquium on Algebraic Geometry was held at the Tata Institute of Fundamental Research, Bombay on 16-23 January, 1968. The Colloquium was a closed meeting of experts and others seri- ously interested in Algebraic Geometry. It was attended by twenty-six members and thirty-two other participants, from France, West Germany, India, Japan, the Netherlands, the Soviet Union, the United Kingdom and the United States. The Colloquium was jointly sponsored, and financially supported, by the International Mathematical Union, the Sir Dorabji Tata Trust and the Tata Institute of Fundamental Research. An Organizing Committee consisting of Professor K. G. Ramanathan (Chairman), Professor M. S. Narasimhan, Professor C. S. Seshadri, Professor C. P. Ramanujam, Pro- fessor M. F. Atiyah and Professor A. Grothendieck was in charge of the scientific programme. Professors Atiyah and Grothendieck represented the International Mathematical Union on the Organizing Committee. The purpose of the Colloquium was to discuss recent developments in Algebraic Geometry. The following twenty mathematicians accepted invitations to ad- dress the Colloquium: S. S. Abhyankar, M. Artin, B. J. Birch, A. Borel, J. W. S. Cassels, B. M. Dwork, P. A. Griffiths, A Grothendieck, F. Hirze- bruch, J.-I. Igusa, Yu. I. Manin, T. Matsusaka, D. Mumford, M. Nagata, M. S. Narasimhan, S. Ramanan, C. S. Seshadri, T. A. Springer, J. L. Verdier and A. Weil. Professor H. Hironaka, who was unable to attend the Colloquium, sent in a paper. iv Report v The Colloquium met in closed sessions. There were nineteen lec- tures in all, each lasting fifty minutes, followed by discussions. Informal lectures and discussions continued during the week, outside the official programme. The social programme included a tea on 15 January, a dinner on 16 January, a programme of classical Indian dances on 17 January, a dinner at the Juhu Hotel on 20 January, an excursion to Elephanta on the morning of 22 January and a farewell dinner the same evening. v Contents Report iv 1 S.S.Abhyankar : Resolution of Singularities of Algebraic Surfaces 1 2 M. Artin : The Implicit Function Theorem in Algebraic Geometry 13 3 B.J.Birch : Diophantine Analysis and Modular Functions 37 4 A.Borel : On the Automorphisms of Certain Subgroups of Semi-Simple Lie Groups 46 5 A.Borel and J. Tits : On “Abstract” Homomorphisms of Simple Alge- braic Groups 78 6 J. W. S. Cassels : Rational Points on Curves of Higher Genus 87 7 B.Dwork : A Deformation Theory for Singular Hyper-surfaces 88 8 P. A. Griffiths : Some Results on Algebraic Cycles on Algebraic Mani- folds 96 9 A. Grothendieck : Standard Conjectures on Algebraic Cycles 205 10 H. Hironaka : Formal Line Bundles along Exceptional Loci 212 11 F. Hirzebruch and K. Janich¨ : Involutions and Singularities 230 vi 12 J-I. Igusa : Geometric and Analytic Methods in the Theory of Theta- Functions 254 13 Yu. I.Manin : On Some Groups Related to Cubic Hyper-surfaces 268 14 T. Matsusaka : On Canonically Polarized Varieties 278 15 D. Mumford : Bi-Extensions of Formal Groups 321 16 M. Nagata : Some Questions on Rational Actions of Groups 338 17 M. S. Narasimhan and S. Ramanan : Vector Bundles on Curves 350 18 C.S.Seshadri : Mumford’s Conjecture for GLp2q and Applications 363 19 T. A. Springer : The Unipotent Variety of a Semi-Simple Group 389 20 J.-L. Verdier : Base Change for Twisted Inverse Image of Coherent Sheaves 408 21 A. Weil : Zeta-Functions and Mellin Transforms 426 vii RESOLUTION OF SINGULARITIES OF ALGEBRAIC SURFACES By Shreeram Shankar Abhyankar 1 1 Introduction. The theorem of resolution of singularities of alge- braic surfaces asserts the following : Surface Resolution. Given a projective algebraic irreducible sur- face Y over a field k, there exists a projective algebraic irreducible non- singular surface Y1 over k together with a birational map of Y1 onto Y (without fundamental points on Y1). For the case when k is the field of complex numbers, after several geometric proofs by the Italians (see Chapter I of [15]), the first rig- orous proof of Surface Resolution was given by Walker [14]. For the case when k is a field of zero characteristic, Surface Resolution was proved by Zariski ([16],[17]); and for the case when k is a perfect field of nonzero characteristic, it was proved by Abhyankar ([2],[3],[4]). A stronger version of Surface Resolution is the following : Embedded Surface Resolution. Let X be a projective algebraic ir- reducible nonsingular three-dimensional variety over a field k, and let Y be an algebraic surface embedded in X. Then there exists a finite se- 1 quence X Ñ X1 Ñ X2 Ñ ... Ñ Xt Ñ X of monoidal transformations, with irreducible nonsingular centers, such that the total transform of Y in X1 has only normal crossings and the proper transform of Y in X1 is nonsingular. For the case when k is of zero characteristic and Y is irreducible, the part of Embedded Surfaces Resolution concerning the proper transform of Y was proposed by Levi [13] and proved by Zariski [18]. Again for the case when k is of zero characteristic, Hironaka [12] proved Embed- ded Resolution for algebraic varieties of any dimension. For the case when k is a perfect field of nonzero characteristic, Embedded Surface Resolution was proved by Abhyankar ([7],[9],[10],[11]). 1 2 S. S. Abhyankar 2 Then in January 1967, in a seminar at Purdue University, I gave a proof of Embedded Surface Resolution (and hence a fortiori also of Surface Resolution) for an arbitrary field k, i.e. without assuming k to the perfect. The details of this proof will be published elsewhere in due course of time. This new proof is actually only a modification of my older proofs cited above. One difference between them is this. In the older proofs I passed to the albegraic closure of k, did resolution there, and then pulled it down to the level of k. In case k is imperfect, the pulling down causes difficulties. In the new modified proof I work directly over k. In this connection, a certain lemma about polynomials in one indeterminate (with coefficients in some field) plays a significant role. In order that my lecture should not get reduced to talking only in terms of generalities, I would like to show you, concretely, the proof of something. The said lemma being quite simple, I shall now state and prove it. 2 The lemma. Let k1 be a field and let q be a positive integer. As- sume that q is a power of the characteristic exponent of k1; recall that, by definition : (the characteristic exponent of k1) = (the characteristic of k1) if k1 is of nonzero characteristic, and (the characteristic exponent of k1) = 1 if k1 is of zero characteristic. Let A be the ring of all polynomials in an indeterminate T with coefficients in k1. As usual, by Aq we denote the set thq : h P Au. Note that then Aq is a subring of A; this is the only property of q which we are going to use. For any h P A, by deg h we shall denote the degree of h in T; we take deg0 “ ´8. Let f P A and let d “ deg f . Assume that f R k, i.e. d ą 0. Let r : A Ñ A{p fAq be the canonical epimorphism. For any nonnegative integer e let Wpq, d, eq “ re{pdqqsq ` rmaxtq ´ pq{dq, pe{dq ´ re{pdqqsqus where the square brackets denote the integral part, i.e. for any real num- ber a, by ras we denote the greatest integer which is ď a. The Lemma. Given any g P A with g R Aq, let e “ deg g. Then we can express g in the form 2 Resolution of Singularities of Algebraic Surfaces 3 g “ g1 ` g˚ f u where g1 P Aq, g˚ P A with g˚ R f A, and u is a nonnegative integer 3 such that u ď Wpq, d, eq and: either u ı 0pqq or rpg˚q R prpAqqq. Before proving the lemma we shall make some preliminary remarks. Remark 1. The assumption that g P A and g R Aq is never satisfied if q “ 1. Thus the lemma has significance only when k1 is of nonzero characteristic and q is a positive power of the characteristic of k1. The lemma could conceivably be generalized by replacing Aq by some other subset of A. Remark 2. For any integers q, d, e with q ą 0, d ą 0, e ě 0, we clearly have Wpq, d, eq ě 0.