Astérisque IGOR DOLGACHEV DAVID ORTLAND Point sets in projective spaces and theta functions Astérisque, tome 165 (1988) <http://www.numdam.org/item?id=AST_1988__165__1_0> © Société mathématique de France, 1988, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 165 ASTÉRISQUE 1988 POINT SETS IN PROJECTIVE SPACES AND THETA FUNCTIONS Igor DOLGACHEV and David ORTLAND SOCIÉTÉ MATHÉMATIQUE DE FRANCE Publié avec le concours du CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE A.M.S. Subjects Classification : 14-02, 14C05, 14C21, 14D25, 14E07, 14H10, 14H40, 14H45, 14J26, 14J25, 14J30, 14K25, 14N99, 14J50, 20C30, 17B20, 17B67. TO HILARY AND NATASHA Table of Contents Introduction 3 I. CROSS-RATIO FUNCTIONS 1. The variety Pmn (first definition) 7 2. Standard monomials .9 3. Examples 13 II. GEOMETRIC INVARIANT THEORY 1. Pmn (second definition) 21 2. A criterion of semi-stability.... 22 3. Most special point sets 2.6 4. Examples 31 III. ASSOCIATED POINT SETS 1. The association 33 2. Geometric properties of associated point sets 41 3. Self-associated point sets .43 IV. BLOWING-UPS OF POINT SETS 1. Infinitely near point sets 53 2. Analysis of stability in m .57 V. GENERALIZED DEL PEZZO VARIETIES 1. The Neron-Severi bilattice 63 2. Geometric markings of gDP-varieties 67 3. The Weyl groups Wn,m 71 4. Discriminant conditions 79 5. Exceptional configurations 81 VI. CREMONA ACTION 1. The Cremona representation of the Weyl group Wn,m 84 2. Explicit formulae. 94 3. Cremona action and association 96 4. Pseudo-automorphisms of gDP-varieties 99 5. Special subvarieties of Pmn 104 1 I. DOLGACHEV, D. ORTLAND VII. EXAMPLES 1. Point sets in P, 110 2. Point sets in P2 (m < 5) 111 3. Cubic surfaces (n=2, m=6) 116 4. Del Pezzo surfaces of degree 2 120 5. Del Pezzo surfaces of degree 1 123 6. Point sets in P3 126 7. Point sets in P4 130 VIII. POINT SETS IN IP1 AND HYPERELLIPTIC CURVES 1. The ta functions 1 3 2 2. Jacobian varieties and theta characteristics 137 3. Hyperelliptic curves 141 4. Theta characteristics on hyperelliptic curves 146 5. Thomae's theorem 1 5 3 6. Elliptic curves 155 7. Abelian surfaces 156 IX. CURVES OF GENUS 3 1. Level 2 structures on the Jacobian variety of a curve of genus 3 160 2. Aronhold sets of bitangents to a quartic plane curve 165 3. The varieties S8 and m3(2) 173 4. Theta structures 177 5. Kummer-Wirtinger varieties 1 82 6. Cayley dianode surfaces 187 7. Göpel functions 190 8. Final remaries 1 9 5 Bibliography 1 97 Notations -203 Index 206 Resume • 209 2 Introduction. The purpose of these notes is to re-introduce some of the worK of A. Coble [Co 11 in a language that a modern mathematician can easily understand. There is a well-Known relationship between the theory of invariants of finite sets of points in a projective line and the theory of hyperelliptic curves. The DOOK of Coble gives an account of the theory which generalizes this relationship to point sets in projective spaces of higher dimension and non-hyperelliptic curves. Though some aspects of this theory were Known before Coble (see for example [Fr1l, IFr21, EKal, [Sen 1, [Sen 2], [Sen 31), his exposition is by far the most complete and conceptually motivated. In recent years the booK of Coble was saved from oblivion and the number of references to it grew substantially. This prompted us to serve the mathematical community by giving a modern account of his theory. The contents of these notes is the following. In Chapters I and II we give a development of the general theory of projective invariants for ordered point sets. We use a presentation of these invariants by certain tableaux, along with the straightening algorithm to describe the structure of the ring they form. Following a now standard approach to the theory of invariants [Mu 11, we construct the moduli spaces P™ for the projective equivalence classes of sets of m ordered points in a n-dimensional projective space IPn and provide a description of the stable and semi-stable ones. A rather complete discussion of the "most special" point sets is given. These are the point sets which are parametrized by the spaces P™ . The chapters conclude with some examples that illustrate how the general techniques worK for specific cases. Note that we are able to discern the structure of the moduli spaces in these examples without too much effort, whereas Coble had to devise rather complicated and ingenious methods to reveal the same information. 3 I. DOLGACHEV, D. ORTLAND Chapter III is concerned with the classical concept of association, which is a form of duality between the spaces P™ and Pm-n_2. It is difficult to trace out the origins of this concept, but it was refined and used extensively by Coble [Co 51. Our approach is to show that association arises from an isomorphism between the coordinate rings of the respective moduli spaces, which is based on the notion of duality between tableaux. In the case m = 2n+2 the notion of association leads to the notion of self-association. We provide a criterion, essentially due to Coble, for a stable set to be self-associated. This condition is closely related to questions of independence of point sets with respect to the linear system of quadrics through them which is extensively studied in modern and classical worKs on algebraic curves. After various geometric properties of associated sets, we prove the rationality of the moduli space Sn that parametrizes projective equivalence classes of ordered self-associated point sets. In Chapter IV we extend the invariant theory of points sets to the case where some of the points are considered to be infinitely near. Following a construction from IK1] we construct the variety parametrizing such point sets, and then consider the extension of the action of the projective linear group on this variety. We use some recent results from [ReM to derive explicit criterion of stability of infinitely near points sets. This allows to construct the spaces P™ which are extensions of the spaces P™, and birational morphisms P™ —• P1^ - In Chapter V we begin to consider point sets from a different point of view. Blowing-up such a set gives a certain rational variety, which we call a generalized Del Pezzo variety. The order on the set equips this variety with an additional structure. This additional structure is interpreted as a certain marKing in the 1- codimensional and 1-dimensional components of the Chow ring of this variety. The varieties P™ can be interpreted as certain moduli varieties of marked generalized Del Pezzo varieties. Here the most interesting part of Coble's theory enters into the discussion. This is the notion of root systems and their Weyl groups. The discovery that Coble was aware of some of these notions even in the case of infinite root systems, a long time before Carton's worK, and it goes without saying, before the wonc of V. Kac and R. Moody, was the main motivation for the first author to study his worK. The theory of Del Pezzo surfaces and surface singularities is Known to have a relationship with this theory. A modern account of this can be found for example in [Ma], [Del, CPU. An earlier exposition of this is due to P. Du Val [DV 1-DV41 who apparently was not aware of Coble's worK. A new result of this chapter is a partial description of roots for certain 4 INTRODUCTION root systems in hyperbolic vector spaces. The notion of roots corresponds to the classical notion of a discriminant condition on a point set which substitutes the condition for a binary form to have a multiple root. In Chapter VI we develop the notion of the Cremona action on the point sets. It was observed by Coble and S. Kantor ( in the case n = 2), and later by P. Du Val [DuV 3], [DuY4] that certain types of Cremona transformations of the projective space act birationally on projective equivalence classes of point sets. More precisely they give a representation of a certain Weyl group Wn m in the group of birational automorphisms of P™. Much effort was applied to give a rigorous exposition of this beautiful theory. The Kernel of the Cremona representation of Wnm can be identified with a subgroup of pseudo- automorphisms (i.e. birational automorphisms which are isomorphisms in codimension 1) of the blowing-up of a point set represented by a generic point of P1^. In the case n = 2, the Kernel is the full automorphism group, and we prove, following Coble, that this group is trivial if m > 9. A modern proof of this result, also based on Coble's ideas, was given in [Gil (m = 9) and [Hirl. In Chapter VII we discuss all special cases where the Weyl group Wn m is finite, and compute the Kernel of the Cremona representation. This leads to a beautiful interpretation of certain elements of the center of the Weyl groups as certain types of Cremona transformations in the projective space. We refer to a recent paper of P. Du Val [DuV 4], where, again without mentioning Coble's worK, a nice account of this is given.