Hurwitz spaces of quadruple coverings of elliptic curves and the modulispace of abelian threefolds 3(1, 1, 4)

✍ Scribed by Vassil Kanev

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 323 KB
• Volume: 278
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310233

✦ Synopsis

Abstract

We prove that the moduli space 𝒜~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~3~(1, 4, 4). The result is based on the study the Hurwitz space ℋ︁~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 ℋ︁^0^~4__,A__~(Y) which parametrizes quadruple coverings π : X → Y with Tschirnhausen modules isomorphic to A^–1^, where A ∈ Pic^n/2^Y, and for which π* : J(Y) → J(X) is injective. This is an analog of the result of Arbarello and Cornalba that the Hurwitz space ℋ︁~4__,n__~(ℙ^1^) is unirational. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)