Cluster pictures for Hitchin fibers of rank two Higgs bundles

TL;DR

This paper develops a cluster picture approach to analyze Hitchin fibers of rank two Higgs bundles, providing criteria for semi-stability and describing the dual graph of minimal models over local fields.

Contribution

It extends cluster picture techniques to non-rational base curves and links these to the p-adic volume variation of Hitchin fibers.

Findings

Semi-stability criterion based on branch locus data.

Description of dual graph of minimal regular model for semi-stable curves.

Analysis of p-adic volume variation of Hitchin fibers.

Abstract

Let $\varphi\colon X\rightarrow Y$ be a degree two Galois cover of smooth curves over a local field $F$ of odd characteristic. Assuming that $Y$ has good reduction, we describe a semi-stability criterion for the curve $X$, using the data of the branch locus of the covering $\varphi$. In the case that $X$ has semi-stable reduction, we describe the dual graph of the minimal regular model of $X$ over $F.$ We do this by adopting the notion of cluster picture defined for hyperelliptic curves for the case where $Y$ is not necessarily a rational curve. Using these results, we describe the variation of the p-adic volume of Hitchin fibers over the semi-stable locus of the moduli space of rank 2 twisted Higgs bundles.