On holomorphic polydifferentials in positive characteristic

TL;DR

This paper investigates the structure of holomorphic polydifferentials on algebraic curves over fields of positive characteristic, providing explicit bases, invariants, and module decompositions for specific automorphism groups.

Contribution

It introduces Boseck invariants, offers elementary methods for module structure analysis, and computes bases for holomorphic polydifferentials in cases with cyclic or elementary abelian automorphism groups.

Findings

Explicit bases for $ ext{Omega}(m)$ in cyclic and elementary abelian cases

Introduction of Boseck invariants for module analysis

Application to tangent space computation of deformation functors

Abstract

In this paper we study the space $\Omega(m)$, of holomorphic $m$-(poly)differentials of a function field of a curve defined over an algebraically closed field of characteristic $p>0$ when $G$ is cyclic or elementary abelian group of order $p^n$; we give bases for each case when the base field is rational, introduce the Boseck invariants and give an elementary approach to the $G$ module structure of $\Omega(m)$ in terms of Boseck invariants. The last computation is achieved without any restriction on the base field in the cyclic case, while in the elementary abelian case it is assumed that the base field is rational. An application to the computation of the tangent space of the deformation functor of curves with automorphisms is given.