Tropicalization of theta characteristics, double covers, and Prym varieties

TL;DR

This paper investigates how theta characteristics on algebraic curves behave under tropicalization, revealing their specialization patterns and defining tropical Prym varieties through double covers.

Contribution

It establishes the specialization relationship between algebraic and tropical theta characteristics and introduces the concept of tropical Prym varieties via double covers.

Findings

Effective tropical theta characteristics are specializations of 2^{g-1} even and odd algebraic theta characteristics.

Each unramified double cover of a tropical curve corresponds to a tropical Prym variety.

The paper formalizes the connection between tropical and algebraic curve invariants.

Abstract

We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of $2^{g-1}$ even theta characteristics and $2^{g-1}$ odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.