Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 5

Geometry of Prym varieties for certain bielliptic curves of genus three and five

Pages: 1739 – 1784

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n5.a5

Authors

Adrian Clingher (Department of Mathematics and Statistics, University of Missouri, St. Louis, Mo., U.S.A.)

Andreas Malmendier (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.; and Department of Mathematics and Statistics, Utah State University, Logan, Ut., U.S.A.)

Tony Shaska (Department of Mathematics and Statistics, Oakland University, Rochester, Michigan, U.S.A.)

Abstract

We construct two pencils of bielliptic curves of genus three and genus five. The first pencil is associated with a general abelian surface with a polarization of type $(1,2)$. The second pencil is related to the first by an unramified double cover, the Prym variety of which is canonically isomorphic to the Jacobian of a very general curve of genus two. Our results are obtained by analyzing suitable elliptic fibrations on the associated Kummer surfaces and rational double covers among them.

Keywords

Kummer surfaces, Prym varieties, isogenies of abelian surfaces

2010 Mathematics Subject Classification

14H40, 14J28

Received 13 August 2021

Received revised 2 September 2021

Accepted 13 September 2021

Published 26 January 2022