Faithful action on the space of global differentials of an algebraic curve

TL;DR

This paper proves that a faithful finite group action on an algebraic curve induces a faithful action on the space of global differentials, with a specific exception involving wild ramification and characteristic 2.

Contribution

It establishes the faithfulness of the induced action on differentials for algebraic curves, identifying a unique exception related to wild ramification in characteristic 2.

Findings

Faithful group actions induce faithful actions on differentials for genus ≥ 2.

The exception occurs only when the action is not tame, the quotient genus is 0, and the characteristic is 2.

The result clarifies the relationship between group actions and differential spaces in algebraic geometry.

Abstract

Given a faithful action of a finite group on an algebraic curve of genus at least 2, we prove that the induced action on the space of global holomorphic differentials is faithful as well, except in the following very special case: the given action is not tame, the genus of the quotient curve is 0 and the characteristic of the base field is 2.