Automorphisms of Curves and Weierstrass semigroups for Harbater-Katz-Gabber covers

TL;DR

This paper investigates the structure of p-group Galois covers of the projective line, focusing on ramification jumps, Weierstrass semigroups, and applications to special curves, providing explicit descriptions and deformation insights.

Contribution

It explicitly determines ramification jumps in terms of pole numbers and explores their relation to Weierstrass semigroups, advancing understanding of Galois covers and curve deformations.

Findings

Explicit formulas for ramification jumps in terms of pole numbers

Applications to maximal curves and curves with large automorphism groups

Analysis of Galois module structure of polydifferentials

Abstract

We study $p$-group Galois covers $X \rightarrow \mathbb{P}^1$ with only one fully ramified point. These covers are important because of the Katz-Gabber compactification of Galois actions on complete local rings. The sequence of ramification jumps is related to the Weierstrass semigroup of the global cover at the stabilized point. We determine explicitly the jumps of the ramification filtrations in terms of pole numbers. We give applications for curves with zero $p$--rank: we focus on maximal curves and curves that admit a big action. Moreover the Galois module structure of polydifferentials is studied and an application to the tangent space of the deformation functor of curves with automorphisms is given.