Deformation of Curves with Automorphisms and representations on Riemann-Roch spaces

TL;DR

This paper investigates how automorphisms of smooth projective curves in positive characteristic deform, linking the problem to matrix representation deformations, with a focus on both equicharacteristic and mixed cases.

Contribution

It reduces the deformation problem of automorphisms of curves to matrix representation deformations, providing new insights into their structure in positive characteristic.

Findings

Deformation problem for automorphisms can be simplified to matrix representation deformations.

Analysis covers both equicharacteristic and mixed characteristic cases.

Provides conditions under which the reduction holds.

Abstract

We study the deformation theory of nonsigular projective curves defined over algebraic closed fields of positive characteristic. We show that under some assumptions the local deformation problem for automorphisms of powerseries can be reduced to a deformation problem for matrix representations. We study both equicharacteristic and mixed deformations in the case of two dimensional representations.