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 ${\mathbb{Z}}_2^r$-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.

To appear in Compositio Mathematica. This version is slightly different

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• Stacks of ramified abelian covers
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• A new example of an algebraic surface with canonical map of degree 16
• Geography of minimal surfaces of general type with ${\mathbb{Z}}_2^2$-actions and the locus of Gorenstein stable surfaces
• Stack of $S_3$-covers
• ${\mathbb{Z}}_3$-actions on Horikawa surfaces