Moduli Of Certain Wild Covers Of Curves

Abstract

A fine moduli space (see Chapter~\ref{secn&t} Definition~\ref{finemdli}) is constructed, for cyclic-by-$\mathsf{p}$ covers of an affine curve over an algebraically closed field $k$ of characteristic $\mathsf{p}\u3e0$. An intersection (see Definition~\ref{M}) of finitely many fine moduli spaces for cyclic-by-$\mathsf{p}$ covers of affine curves gives a moduli space for $\mathsf{p}\u27$-by-$\mathsf{p}$ covers of an affine curve. A local moduli space is also constructed, for cyclic-by-$\mathsf{p}$ covers of $Spec(k((x)))$, which is the same as the global moduli space for cyclic-by-$\mathsf{p}$ covers of $\mathbb{P}^1-\{0\}$ tamely ramified over $\infty$ with the same Galois group. Then it is shown that a restriction morphism (see Lemma~\ref{res mor-2}) is finite with degrees on connected components $\textsf{p}$ powers: There are finitely many deleted points (see Figure 1) of an affine curve from its smooth completion. A cyclic-by-$\mathsf{p}$ cover of an affine curve gives a product of local covers with the same Galois group, of the punctured infinitesimal neighbourhoods of the deleted points. So there is a restriction morphism from the global moduli space to a product of local moduli spaces