Algebraic Geometry: Proceedings, Tromsø Symposium, Norway, June 27 – July 8, 1977 PDF
Preview Algebraic Geometry: Proceedings, Tromsø Symposium, Norway, June 27 – July 8, 1977
Lecture Notes ni Mathematics Edited by .A Dold and .B Eckmann 687 ciarbeglA yrtemoeG Proceedings, Tromse Symposium, Norway, June 27 - July 8, 1977 Edited by Loren .D Olson l l l l k l l I M W I U 1 1 1 1 1 l W 1 1 1 1 1 1 1 1 1 1 1 1 1 I I ETHICS ETH-BIB O0100000362269 galreV-regnirpS Berlin Heidelberg New kroY 8791 Editor Loren .D Olson Mathematics Department of Tromso University N-9001 Tromso/Norway AMS Subject Classifications (1970): 41 C15,14 H 99,14.110,14 A05,14 N 01 ISBN 3-540-08954-3 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-08954-3 Springer-Verlag NewYork Heidelberg Berlin This work si subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, -er printing, re-use of illustrations, broadcasting, reproduction yb photocopying machine or similar means, and storage ni data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed ni Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. PREFACE From June 27 to July 8, 1977, a symposium on algebraic geometry was held at the University of Troms~, Norway. The lectures delivered there were primarily focused on the two topics of intersection theory and space curves. The contributions presented here are in large measure based on talks given at the symposium. The symposium was supported financially by the Norwegian Research Council for Science and Humanities (NAVF) and by the University of Troms~. We wish to express our thanks to both for their support. Loren D. Olson TABLE OF CONTENTS WILLIAM FULTON AND ROBERT MACPHERSON, Defining Algebraic Intersections ............... LAURENT GRUSON ET CHRISTIAN PESKINE, Genre des courbes de l'espace projectif ....... .. I13 AUDUN HOLME, Deformation and Stratification of Secant Structure ...................................... 6O BIRGER IVERSEN, Depth Inequalities for Complexes ............... 92 O.A. LAUDAL, A Generalized Trisecant Lemma .................. 112 O.A. LAUDAL AND K. L~NSTED, Deformations of Curves I. Moduli for Hyperelliptic Curves ........................... 150 ULF PERSSON, Double Coverings and Surfaces of General Type .. 168 RAGNI PIENE, Some Formulas for a Surface in i ~ 3 196 i i o i i b i I I I I I I MARGUERITE FLEXOR ET LUCIEN SZPIRO, Un th~or~me de structure locale pour les complexes parfaits ............................. 236 DEFINING ALGEBRAIC INTERSECTIONS William Fulton and Robert MacPherson (Brown University, Providence, RI) Contents .i When is V.W uniquely defined? 2. How well defined is V°W in general? .3 The deformation .4 Application to proper intersections .5 Cone bundles and Segre classes .6 Continuity 7. A classical enumeration problem Suppose V and W are subvarieties of dimension v and w of a nonsingular algebraic variety X of dimension n . We are going to study the following: Intersection Problem Find an equivalence class V.W of algebraic v + w - n cycles which represents the algebraic intersection of V and W . The traditional approach to this problem is to solve it directly only for the particular case where V and W meet so as to deter- mine a unique intersection cycle. Then for general cycles V and W , the problem is solved by first moving one of them in a rational family so that this particular case applies. Since the motion in- volved is not uniaue, the intersection cycle so constructed is not uniquely determined. However, it is defined up to rational eauiva- lence in X by this procedure. Our object here is to consider the intersection problem directly for arbitrary V and W without moving them first. The aim is to define V.W within as small an eauivalence class as possible. i. When is V.W uniquely defined? In this section, as background, we review the traditional ap- proach to the intersection problem: to give sufficient conditions that V-W be a uniquely defined cycle, and to construct it under those conditions. We also review the analogous situation in topology and analysis. Geometry V and W are said to intersect properly if each irreducible component .Z of V n W has dimension v + w - n If V and W 1 intersect properly, then V-W is uniauely determined as a linear combination ~m(Zl)Z i of the components Z i . In this case the intersection problem becomes one of determining the coef- ficients m(Z )i , called the intersection multiplicities. Any subscheme S of X of pure dimension i determines a canonical algebraic cycle S : if .Z are the irreducible compo- 1 nents of S and .z are their generic points, 1 S = length ~ (~S)z.~Z" 1 z i 1 If V and W intersect properly, a naive guess would be that V.W = IV n W , i.e. that m(Z )i = length l This formula is adequate for curves on a surface (or, in fact, for the case that V or W is a divisor). But, as is well known, it is too large in general. (See ii for a discussion of its failure and the history of the development of a correct formula by Severi, Well, Chevalley, and Samuel.) Serre gave an elegant formula for the inter- section multiplicities with correction terms to the naive guess: m(Z )i = length ~z. I ~v ~X ~W)zi 1 + ~ (-i) jlength ,O z or ~x j=l i j (~V' z i In a sense, this formula explains why the naive guess fails. For another explanation, see §4. Topology In topology, the intersection problem is to find an intersection cycle V-W where V and W are, say, integral simplicial v- and w- cycles on an oriented polyhedral n-manifold X . This problem was studied when v + w = n by Poincar4 10 and for arbitrary dimen- sions by Lefschetz 9. The cycles V and W are said to be dimensionally transverse if every simplex in V n W has dimension < v + w. - n . If V and w are dimensionally transverse then they determine a unique inter- section cycle m(~)a where the sum is over all v + w - n dimen- sional simplices ~ (supplied with orientations) in V n W , and m(a) is called the intersection multiplicitv of V and W at ~ . The intersection multiplicity m(~) may be determined by the following procedure which is equivalent to that of Lefschetz. Choose a local coordinate chart ~n ~ U ÷ X near ~ compatible with the orientation of X . Then there is a map ~ : (V n U) x (W n U) ÷ × ~n that sends x,y to p(x) ,y - x where p(x) in the or- thogonal projection of x on s and y- x is the vector differ- ence in ~n . The intersection multipl~city m(~) then is the lo- cal degree of ~ near (b,0) e ~ × RI n where b is an interior point to s . As in geometry, if V is not dimensionally transverse to W V is traditionally first deformed to V' which is dimensionally transverse to W . The class of the cycle V' • W in Hv+w_ n )X( is independent of the deformation chosen. Thus for general V and W , V.W is defined up to homology in X . An_alysis In analysis, the intersection problem is given two currents ~i and ~2 on a compact manifold X , to find a product that cor- responds to their homology product. If ~i and ~2 are both smooth differential forms, then the product is just the wedge product. De Rham in his thesis i showed that this corresponds to the Lefschetz intersection product (seven years before the invention of the cohomologv cup product). More generally if the singular supports of ~I and ~2 are disjoint, or more generally still if their wave front sets are disjoint, then a unique product exists 6. If the currents arise from a pair of al- gebraic varieties intersecting properly, then the product can be de- termined by geometric measure theoretic methods 7. But no general construction of a uniaue product encompassing all these cases exists. Perhaps the general problem is best stated this way: There exist families H t of smoothing operators converging to the identity as t ÷ 0 but there is no canonical choice. When is lim Ht~ 1 ^ Ht~ 2 t÷0 independent of H t ? (Or more generally, characterize the set of possible limits. 2. How well defined is V.W in general? Suppose v n W has components of dimension greater than v + w - n . Then how well defined can we expect the intersection cycle ,V W to be? First, it is too much to expect that it should be uniauely de- fined. For consider the case where X is the projective plane and V and W are both the same projective line contained in X . By Bezout's theorem we know that the algebraic intersection consists of a point with multiplicity one. However, there is no distinguished point in the given configuration of V, W, and X . In fact, the or- bits of the symmetry group of the given data are W V n and X - V n W . So no better answer exists than the class consisting of any point on V n W . (Another reason it is too much to expect a uniquelv defined in- tersection product comes from analog v with the topological situation. According to a principle which is widely believed but not precisely formulated or proved, a globally defined commutative product on the chain level that gives the usual product for H, (X~) cannot exist. A chain level product can be globally defined but not commutative, as in the cohomology cup product, or commutative but not aloballv de- fined, as in the homology intersection product. A globall v defined commutative product would presumablv contradict the existence of co- homology operations. The original construction of the Steenrod squares, for example, explicitly used the non-commutativity of the chain level cup product. Note that over the rationals, where no co- homology operations exist, such a product can exist and was construc- ted by Sullivan [14]. In §3, we will construct a product V.W which satisfies the following. Assertion 1 v.w is well defined in Av+w_ n V( n W) . That ,si the algebraic intersection V.W lies in the physical inter- section V n W and is well defined up to rational eauivalence in V n W . Assertion 1 agrees with the traditional idea described in §I that the intersection cycle is uniauely defined when V and W in- tersect properly. This is because the homomorphism of the group of algebraic v + w- n cycles on V n W to Av+w_ n(v n W) is an iso- morphism if and only if V intersects W properly. So for proper intersections, a well defined class in Av+w_ n (V n W) is a uniauely determined algebraic cycle. Assertion 1 also agrees with the above example where V = W = a projective line in the projective plane since the set of points in V n W is exactly the set of effective cycles in a class of A 0(v n W) . Another reason to expect Assertion 1 is that the corresponding topological statement is true. This can be seen by a slight refine- ment of the usual transversality techniaues. For suppose V and W are cycles in a piecewise linear manifold X . Let N be a regu- lar neighborhood of V n W (so N deformation retracts to V n W ). Now V can be deformed slightly to V' so that all mo- tion takes place strictly inside N and V' is dimensionally trans- verse to W . V V' Diagram 1
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