# Moduli of certain wild covers of curves

## TL;DR

This paper constructs and analyzes fine moduli spaces for certain wild covers of affine curves in characteristic p, revealing their structure, local-global relations, and finiteness properties of restriction morphisms.

## Contribution

It introduces new fine moduli spaces for cyclic-by-p covers of affine curves and establishes their local-global relationships and finiteness of restriction maps.

## Key findings

- Constructed fine moduli spaces for cyclic-by-p covers.
- Established a finite restriction morphism with p-power degrees.
- Linked global and local moduli spaces via product structures.

## Abstract

A fine moduli space is constructed, for cyclic-by-$\mathsf{p}$ covers of an affine curve over an algebraically closed field $k$ of characteristic $\mathsf{p}>0$. An intersection of finitely many fine moduli spaces for cyclic-by-$\mathsf{p}$ covers of affine curves gives a moduli space for $\mathsf{p}'$-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 is finite with degrees on connected components $\mathsf{p}$ powers: There are finitely many deleted points 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.