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The unirationality of \({\mathcal A}_ 5\). (English) Zbl 0589.14043

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 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.
Reviewer: M.Miyaniski

MSC:

14M20 Rational and unirational varieties
14K10 Algebraic moduli of abelian varieties, classification
14C21 Pencils, nets, webs in algebraic geometry
14K25 Theta functions and abelian varieties
14H10 Families, moduli of curves (algebraic)
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