Vol. 15, No. 3, 2021

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Skeletons of Prym varieties and Brill–Noether theory

Yoav Len and Martin Ulirsch

Vol. 15 (2021), No. 3, 785–820
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.

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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