Hurwitz spaces of quadruple coverings of elliptic curves and the moduli space of abelian threefolds A{sub}3(1,1,4)

摘要

We prove that the moduli space A{sub}3(1,1,4) of polarized abelian threefolds with polarization of type (1,1,4) is unirational. By a result of Birkenhake and Lange this implies the unirationality of the isomorphic moduli space A{sub}3(1,1,4). The result is based on the study the Hurwitz space H{sub}(4,n)(Y) of quadruple coverings of an elliptic curve Y simply branched in n ≥ 2 points. We prove the unirationality of its codimension one subvariety (H{sub}(4,A)){sup}0(Y) which parametrizes quadruple coverings π: X → Y with Tschirnhausen modules isomorphic to A{sup}(-1), where A ∈ Pic{sup}(n/2)Y, and for which π{sup}*: J(Y) → J(X) is injective. This is an analog of the result of Arbarello and Cornalba that the Hurwitz space H{sub}(4,n)(P{sup}1) is unirational.