Equiramified deformations of covers in positive characteristic

TL;DR

This paper investigates the deformation space of wildly ramified covers of curves in positive characteristic, providing explicit bounds on the dimension of the moduli space and demonstrating the bounds are sharp for abelian p-group covers.

Contribution

It explicitly constructs the moduli space of equiramified deformations and derives bounds on its dimension based solely on the ramification filtration, with sharpness shown in special cases.

Findings

The moduli space $M_\phi$ is a subscheme of an explicitly constructed scheme.

Explicit upper and lower bounds for the dimension $d_\phi$ depend only on the ramification filtration.

For abelian p-group covers, the upper bound for $d_\phi$ is achieved.

Abstract

Suppose $\phi$ is a wildly ramified cover of germs of curves defined over an algebraically closed field of characteristic p. We study unobstructed deformations of $\phi$ in equal characteristic, which are equiramified in that the branch locus is constant and the ramification filtration is fixed. We show that the moduli space $M_\phi$ parametrizing equiramified deformations of $\phi$ is a subscheme of an explicitly constructed scheme. This allows us to give an explicit upper and lower bound for the Krull dimension $d_\phi$ of $M_\phi$. These bounds depend only on the ramification filtration of $\phi$. When $\phi$ is an abelian p-group cover, we use class field theory to show that the upper bound for $d_\phi$ is realized.