Theta characteristics of hyperelliptic graphs

TL;DR

This paper investigates the relationship between theta characteristics of hyperelliptic metric graphs and their associated hyperelliptic curves over non-Archimedean fields, revealing how curve theta characteristics specialize to graph theta characteristics.

Contribution

It establishes a detailed correspondence between theta characteristics on hyperelliptic graphs and their algebraic curve counterparts, including specialization counts for even and odd characteristics.

Findings

Each effective graph theta characteristic corresponds to 2^{g-1} even and 2^{g-1} odd curve theta characteristics.

All non-effective graph theta characteristics on the graph correspond to 2^{g} even curve theta characteristics.

The study extends the understanding of tropical and algebraic theta characteristics in hyperelliptic settings.

Abstract

We study theta characteristics of hyperelliptic metric graphs of genus $g$ with no bridge edges. These graphs have a harmonic morphism of degree two to a metric tree that can be lifted to morphism of degree two of a hyperelliptic curve $X$ over $K$ to the projective line, with $K$ an algebraically closed field of char$(K) \not =2$, complete with respect to a non-Archimedean valuation, with residue field $k$ of char$(k)\not=2$. The hyperelliptic curve has $2^{2g}$ theta characteristics. We show that for each effective theta characteristics on the graph, $2^{g-1}$ even and $2^{g-1}$ odd theta characteristics on the curve specialize to it; and $2^g$ even theta characteristics on the curve specialize to the unique not effective theta characteristics on the graph.