Non-normal abelian covers

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.