Hurwitz spaces of quadruple coverings of elliptic curves and the modulispace of abelian threefolds 𝒜3(1, 1, 4)

2003/08/01 by Vassil Kanev
Mathematics · #Abelian group #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Codimension #Combinatorics #Elliptic curve #Geometry and complex manifolds #Injective function #Mathematics #Moduli space #Pure mathematics #Space (punctuation) #Subvariety #math.AG #msc:14H10 #msc:14H30 #msc:14K10

paper · pdf · doi:10.1002/mana.200310233

published as Math. Nachr. Vol. 278 (2005), pp. 154 - 172. · 28 pages, amslatex, to appear in Mathematische Nachrichten

arxiv created 2003/08/01 · openalex publication_date 2004/12/17 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

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)

Citations

• A note on Hurwitz schemes of covers of a positive genus curve
• Vector Bundles over Elliptic Fibrations
• Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of Abelian threefolds
• Vector Bundles Over an Elliptic Curve
• Algebraic Geometry