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Lecture Notes in Mathematics Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 1196 Eduardo Casas-Alvero Sebastian Xamb6-Descamps The Enumerative Theory of Conics after Halphen ! II Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Authors Eduardo Casas-Alvero Sebastian Xamb6-Descamps Department of Geometry and Topology, University of Barcelona Gran Via 585, Barcelona 08007, Spain Mathematics Subject Classification (1980): 14N 10; 14C 17; 51 N 15 ISBN 3-540-16495-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-16495-2 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payabie to "Verwertungsgesellschaft Wort", Munich. © Springer-VerlagBerlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210 Introduction. This work deals on the one hand with understanding the contents of Halphen ' s contribution to the subject of enumerative theory of conics, and on the other with extending his theory to conditions of any codimension. The reader interested in the history of this subject may profit from the beautiful paper of Kleiman [K.2 ]. In the enumerative theory of conics there have been basically three approaches, namely those associated to De Jonqui~res, to Chasles, and to Ha]phen /see the works of these authors referred to in the references, as well as [K.2 ] and the references thereinJ. Conceptually the first two are similar in that they correspond to computations performed in the Chow ring of P5 and of the variety of complete conics, respectively. Unfortunately the numbers obtained with these aproaches : need not have enumerative significance, even if the data in the problem under consideration are in general position. A famous example of this failure is the answer given by De Jonquibres theory to the problem of finding the number of conics that are tangent to five given conics in general position. Similarly, Halphen gave examples of this unsatis- factory situation, needless to say a little more involved, for the theory of Chasles (see [H.3.], §15, or the example 1~.8 in this memoir]. On the other hand, the starting point in Halphen's theory is the distin- tion between proper and improper solutions /see §bJ to an enumerative problem and his goal is to count the number of proper solutions. The numbers produced with this theory have always enumerative significance in the sense that if the data of the /reduced] conditions involved are in general position, then such numbers always are the number of distinct proper solutions of the problem. In addition, it turns out that all nondegenerate solutions are proper solutions and, if the data of the conditions are in general position, then, conversely, all proper solutions are non-degenerate, so that for (reduced] conditions with data in general position Halphen 's theory gives the number of non-degenerate solutions. In relation to this last point we should say that recently De Concini and Procesi /D-P] have taken the number of non-degenerate solutions, in the general setting of symmetric spaces, as the corner stone for an abstract enume- rative theory. IV The present work is the result of a project begun about two years ago by the first author with the idea of understanding Halphen's results and of providing modern proofs for them. This took about one year and the output was roughly the contents of §§ 1-1~. Halphen considered two kinds of enume- rative problems, namely, (1) to find the number of conics in a one-dimensional system that properly satisfy a given first order condition, and (2) to find the number of conics properly satisfying five independent conditions. These problems are solved by what we call ttalphen's first and second formula, which are the contents of Theorem 9.2 and 1~.6. Although the basic ideas of this first part are due to Halphen, the pre- sentation and many of the proofs are new. This is especially so for the defini- tion of local characteristic numbers of first order conditions and the proof of Halphen's first formula. After this first part had taken shape we became interested in finding analogues to these ideas for conditions of any codimension. The joint work in this direction has been developed in the last twelve months and the results are the contents of §§ 15-23. In spite of the fact that the results of the first part can be obtained again from results of the second, we have nevertheless maintained the two parts in order to offer, in the first, a rather elementary and updated version of Halphen's work on the subject, and, in the second, a general treatment for conditions of any codimension. Finally we give a brief description of the contents of §§ 15-23. Section 15 is devoted to recall the structure of the Chow ring of the variey W of complete conics and to list a number of cycles and relations among them which are needed later on. Sections 16, 17 and 18 are more general than the rest and are devoted to prove a generalization of the classical formula of Noether about the intersec- tion of plane curves (see theorem 16.6) and to use it for a generalization of Halphen's first formula (see theorem 18.5). Sections 19 and 20 are devoted to the construction of certain conditions (cycles), to the definition of strict equivalence of conditions and the groups Hal" (W), and to prove that the strict equivalence classes of those cycles provide a free 7Z-basis for Hal'(W). The main tool here is a particularization of Halphen's generalized formula (18.5) to the case of conics and a numerical criterion for strict equivalence proved with the resulting formula (20.2 and theorem 20.4). In section 2I the graded group Hal'(W) is ~iven a structure of graded commutative ring with unit, the enumerative significance of which is explained in theorem 21.7. This ring is the abstract ring of De Concini and Procesi in the case of conics. The product in Hal'(W) is made explicit in section 22 by showing how to compute the products of any two terms of the basis construc- ted before. In section 23 we work out two examples. E. Casas Alvero S. )Camb6 Descamps List of notations and conventions g ring of integers Q field of rational numbers field of real numbers C field of complex numbers R* group of units of the ring R P n-dimensional projective space over C n n dual space of Pn' or projective hyperplane space ord (f) least exponent of the non-zero terms of the (broken) power series f X ~'x,z local ring of the irreducible subvariety Z of an algebraic variety X If Z c_ y c_X, Y defines an ideal in GX, Z that we shall call "ideal of Y in ~. " X,Z " Sing(X) the singular set of an algebraic variety X D~D ' divisors D and D' are linearly equivalent o (K), K ° the result of transforming K by an element o of a group acting on a set of objects where K belongs. Given cycles K and K' on a smooth algebraic vari¢ty X, and a component Z of IKInIK'I, where [K[ means the support of K, if dim(Z) = dim(K)+ +dim(K' )-dim(X), then Z will be called a proper component of the inter- section K n K' and the intersection multiplicity of K and K' at Z will be denoted by iz(K'K'). The variety X will be clear from the context. If K and K' meet properly, i.e., all components of K~K' are proper, then K.K' will denote their intersection cycle, i.e., Eiz(K'K')'Z, the summation extended over all components of the intersection. Given a zero cycle K on X, and a subset V of X, #V K will mean the sum of the multip.licities-£ in K of the points of V. Instead of #X K we will also write IX K or deg(K). If X is complete and a is a VIII ( rational class of O-cycles, IX ~ will denote the degree for any cycle repre- senting c, . Suppose K and K' are such that dim(K)+dim(K') = dim(X) and that K and K' intersect properly. Then X (K'K') will also be denoted simply by (K.K'), or K.K' if no confusion should arise, and will be called (total) intersection number of K and K'. A" (X) will denote the intersection ring of X, graded by codimension. The class of a cycle K in A" (X) will be denoted by [K]. I~ X and X' are smooth projective varieties and f: X----+ X' is a morphism, f,: A'(X) > A'(X') and f*: A" (X') ,A" (X) will denote the usual push-forward and pull-back maps associated to f. CONTENTS Introduction, List of notations and conventions, §i Conics, p. 1 §2 Complete conics, p. 3 §3 Systems of conics, p. 8 §4 Equations of the degeneration hypersurfaces, p. ii §5 Conditions imposed on conics, p. 14 §6 Action of the projective group and proper solutions of an enumerative problem, p. 18 §7 Genericity of the data of conditions, p. 22 §8 Local characters of a condition, p.
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