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Hurwitz spaces of quadruple coverings of elliptic curves and the modulispace of abelian threefolds 𝒜 3 (1, 1, 4)
Author(s) -
Kanev Vassil
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310233
Subject(s) - mathematics , moduli space , subvariety , abelian group , pure mathematics , codimension , injective function , elliptic curve , space (punctuation) , combinatorics , linguistics , statistics , philosophy , variety (cybernetics)
We prove that the moduli space 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 3 (1, 4, 4). The result is based on the study the Hurwitz space ℋ 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 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 ℋ 4 ,n (ℙ 1 ) is unirational. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)