Quadratic differentials and equivariant deformation theory of curves

TL;DR

This paper investigates the deformation theory of algebraic curves with finite p-group actions, linking tangent space dimensions to coinvariant spaces of quadratic differentials, and computes these in specific cases.

Contribution

It establishes a precise relationship between deformation tangent spaces and coinvariants, and computes these dimensions for cyclic and weakly ramified group actions.

Findings

Dimension of tangent space equals coinvariants of G on quadratic differentials

Explicit calculations for cyclic p-groups and weakly ramified actions

Identification of p-rank subrepresentations of the differential space

Abstract

Given a finite p-group G acting on a smooth projective curve X over an algebraically closed field k of characteristic p, the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of G acting on the space V of global holomorphic quadratic differentials on X. We apply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when G is cyclic or when the action of G on X is weakly ramified. Moreover we determine certain subrepresentations of V, called p-rank representations.