A descent map for curves with totally degenerate semi-stable reduction

TL;DR

This paper develops a method to compute torsion subgroups and divisibility properties of line bundles on curves with totally degenerate semi-stable reduction over local fields, advancing understanding of their arithmetic structure.

Contribution

It introduces a descent map for such curves, enabling explicit calculations of torsion points and line bundle divisibility, which were previously difficult to determine.

Findings

Computed the prime-to-p rational torsion subgroup of the Jacobian.

Determined divisibility of line bundles, including theta characteristics.

Analyzed rationality and higher spin structures on the curve.

Abstract

Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to-$p$ rational torsion subgroup on the Jacobian of $C$. We also determine divisibility of line bundles on $C$, including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of $C$.