Maximal monodromy in unequal characteristic

TL;DR

This paper investigates the monodromy properties of $p$-cyclic covers over mixed characteristic discrete valuation rings, including monodromy extension, group, filtration, and Swan conductor, with specific focus on hyperelliptic curves of genus 2 when p=2.

Contribution

It provides a detailed analysis of monodromy invariants for special covers of high genus and hyperelliptic curves in mixed characteristic settings, extending understanding of their stable models.

Findings

Determined monodromy extension and group for high genus covers

Computed Swan conductor and filtration in the monodromy group

Analyzed hyperelliptic curves of genus 2 for p=2 case

Abstract

Let $R$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with fraction field $K$. We study stable models of $p$-cyclic covers of $\Proj_K$. First, we determine the monodromy extension, the monodromy group, its filtration and the Swan conductor for special covers of arbitrarily high genus with potential good reduction. In the case $p=2$ we consider hyperelliptic curves of genus 2.