Bielliptic curves of genus three and the Torelli problem for certain elliptic surfaces

2017/01/24 by Atsushi Ikeda, Ikeda, Atsushi · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1701.06984

Abstract

We study the Hodge structure of elliptic surfaces which are canonically defined from bielliptic curves of genus three. We prove that the period map for the second cohomology has one dimensional fibers, and the period map for the total cohomology is of degree twelve, and moreover, by adding the information of the Hodge structure of the canonical divisor, we prove a generic Torelli theorem for these elliptic surfaces. Finally, we give explicit examples of the pair of non-isomorphic elliptic surfaces which have the same Hodge structure on themselves and the same Hodge structure on their canonical divisors.

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• A non-linear differential equation for the periods of elliptic surfaces

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