Non-normal abelian covers

TL;DR

This paper investigates non-normal abelian covers of varieties, focusing on cases where the varieties are $S_2$ with mild singularities, and applies these results to construct compactifications of moduli spaces of surfaces.

Contribution

It extends the study of abelian covers to non-normal varieties with specific singularity conditions, especially $ ext{Z}_2^r$-covers of surfaces, providing new tools for moduli space compactifications.

Findings

Analyzed non-normal abelian covers with mild singularities.

Developed methods for $ ext{Z}_2^r$-covers of surfaces.

Applied results to moduli space compactifications.

Abstract

An abelian cover is a finite morphism $X\to Y$ of varieties which is the quotient map for a generically faithful action of a finite abelian group $G$. Abelian covers with $Y$ smooth and $X$ normal were studied in \cite{Pardini_AbelianCovers}. Here we study the non-normal case, assuming that $X$ and $Y$ are $S_2$ varieties that have at worst normal crossings outside a subset of codimension $\ge 2$. Special attention is paid to the case of $\Z_2^r$-covers of surfaces, which is used in arxiv:0901.4431 to construct explicitly compactifications of some components of the moduli space of surfaces of general type.