Equiramified deformations of covers in positive characteristic

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.