Automorphisms of curves and Weierstrass semigroups for Harbater–Katz–Gabber covers

Abstract

We study p p -group Galois covers X → P 1 X \rightarrow \mathbb {P}^1 with only one fully ramified point in characteristic p > 0 p>0 . These covers are important because of the Harbater–Katz–Gabber compactification theorem 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 p -rank: we focus on curves that admit a big action. Moreover, we initiate the study of the Galois module structure of polydifferentials.