Skeletons of Prym varieties and Brill–Noether theory

Len, Yoav; Ulirsch, Martin

doi:10.2140/ant.2021.15.785

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Skeletons of Prym varieties and Brill–Noether theory

Yoav Len and Martin Ulirsch

Vol. 15 (2021), No. 3, 785–820

DOI: 10.2140/ant.2021.15.785

Abstract

We show that the non-Archimedean skeleton of the Prym variety associated to an unramified double cover of an algebraic curve is naturally isomorphic (as a principally polarized tropical abelian variety) to the tropical Prym variety of the associated tropical double cover. This confirms a conjecture by Jensen and the first author. We prove a new upper bound on the dimension of the Prym–Brill–Noether locus for a generic unramified double cover in a dense open subset in the moduli space of unramified double covers of curves with fixed even gonality on the base. Our methods also give a new proof of the classical Prym–Brill–Noether theorem for generic unramified double covers that is originally due to Welters and Bertram.

Keywords

tropical Prym variety, non-Archimedean uniformization, folded chain of loops, Prym–Brill–Noether locus

Mathematical Subject Classification

Primary: 14H40, 14Txx

Milestones

Received: 7 May 2020

Revised: 27 August 2020

Accepted: 10 October 2020

Published: 20 May 2021

Authors

Yoav Len
Mathematical Institute University of St. Andrews St. Andrews United Kingdom

Martin Ulirsch
Institut für Mathematik Goethe-Universität Frankfurt Frankfurt am Main Germany

Vol. 15, No. 3, 2021