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Point sets in projective spaces and theta functions. (English) Zbl 0685.14029

Astérisque, 165. Paris: Société Mathématique de France; Centre National de la Recherche Scientifique. 210 p. FF 145.00; $ 24.00 (1988).
The book under review is devoted to the restauration (in the frame of modern algebraic geometry) of some beautiful classical work mainly due to A. B. Coble [cf. Coble’s book: “Algebraic geometry and theta functions” (New York 1929)].
The main objects of this work are the moduli spaces \({\mathbb{P}}^ m_ n\) for the projective equivalence classes of ordered m-uples of points in an n-dimensional projective space \({\mathbb{P}}_ n\). One of the most exciting aspects concerning these moduli spaces is their relation with root systems (relation discovered by Coble, Kantor and Du Val), specifically the existence of a representation of a certain Weyl group \(W_{n,m}\) in the group of birational automorphisms of \(P^ m_ n\); this topic is discussed in the chapters V-VII of the book.
The last two chapters are devoted to the link between the spaces \(P^ m_ n\) and abelian varieties. The main (classical) observation is that an ordered set of \(2g+2\) points in \({\mathbb{P}}_ 1\) defines a hyperelliptic curve of genus g together with a level 2 structure on its Jacobian and similarly an ordered set of 7 points in \({\mathbb{P}}_ 2\) defines a curve of genus 3 together with a level 2 structure. It is shown for instance that there is a birational morphism from the moduli space \(M_ 3(2)\) of curves of genus 3 with level 2 structure to the space \({\mathbb{P}}^ 7_ 2\) defined (roughly speaking) by associating to each curve C in \(M_ 3(2)\) a “geometrically marked” Del Pezzo surface S of degree 1 which is a double cover of \({\mathbb{P}}_ 2\) branched along C (identified in an obvious way with a point in \({\mathbb{P}}^ 7_ 2)\).
Reviewer: A.Buium

MSC:

14N05 Projective techniques in algebraic geometry
14D20 Algebraic moduli problems, moduli of vector bundles
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14K25 Theta functions and abelian varieties
14E99 Birational geometry