The unirationality of \({\mathcal A}_ 5\).
type: Item
instance of: scholarly article
title: The unirationality of \({\mathcal A}_ 5\).
zbMATH Open document ID: 0589.14043
DOI: 10.2307/2007041
published in: Annals of Mathematics. Second Series
publication date: 1984-01-01T00:00:00Z
review text: Let \({\mathcal A}_ 5\) be the moduli space of 5-dimensional, principally polarized abelian varieties and let \({\mathcal R}_ 6\) be the Prym moduli space parametrizing double covers of curves of genus 6. By \textit{W. Wirtinger} [''Untersuchungen über Thetafunctionen'' Teubner, Leipzig 1895), there is the Prym map \(p : {\mathcal R}_ 6\to {\mathcal A}_ 5\) assigning to a double cover \(\pi : \tilde C\to C\) the neutral component of \(Ker(\pi_* : J(\tilde C)\to J(C)),\) where \(J(\;)\) denotes the Jacobian variety, and p is known to be dominant. A main result of the present article is to show that \({\mathcal R}_ 6\) and \({\mathcal A}_ 5\) are unirational. The idea is to use the theory of nets of quadrics. A net of quadrics in \({\mathfrak P}^ 6\) is a family of quadrics \(\{\) A(t);t\(\in \Pi \}\) in \({\mathfrak P}^ 6\) parametrized by a projective plane \(\Pi\). Such a net is invertible if a general member is nondegenerate and if every member has rank\(\geq 6\). Associated to such a net, we have a pair (C,L) of the discriminant locus \(C=\{t\in \Pi;\;A(t) \text{is degenerate}\}\), i.e., the locus of singular quadrics, and a non-vanishing theta characteristic L. Let \(\bar N_ 0\) be the space of projective equivalence classes of invertible nets whose discriminant locus is \(S\cup \ell\), where S is a sextic in \({\mathfrak P}^ 2\) with 4 double points and \(\ell\) is a line. Then there is a dominant map \(f : \bar N_ 0\to {\mathcal R}_ 6\) via the discriminant map \(\{A(t);t\in \Pi \}\mapsto (S\cup \ell,L)\mapsto (\nu (S),\pi),\) where \(\nu\) (S) is the normalization of S and \(\pi\) is an étale double cover of \(\nu\) (S) given by a line bundle \(L\otimes {\mathcal O}(-2)\). More precisely, f is dominant on each component of \(\bar N_ 0\). By a detailed analysis of \(\bar N_ 0\), the author shows that \({\mathcal R}_ 6\) and \({\mathcal A}_ 5\) are unirational. We note that the unirationality of \({\mathcal A}_ 4\) was proved by H. Clemens.
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 14C21
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 3946349
zbMATH Keywords: unirationality of moduli space of 5-dimensional principally
zbMATH Keywords: polarized abelian varieties
zbMATH Keywords: Prym moduli space parametrizing double covers of curves of
zbMATH Keywords: genus 6
zbMATH Keywords: nets of quadrics
zbMATH Keywords: theta characteristic
zbMATH Keywords: Prym moduli space parametrizing double covers of curves of genus 6
author: Ron Donagi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2007041
OpenAlex ID: W1980682176
description: scientific article; zbMATH DE number 3946349