Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

Abstract

Given a smooth, projective curve $Y$, a finite group $G$ and a positive integer $n$ we study smooth, proper families $X\to Y\times S\to S$ of Galois covers of $Y$ with Galois group isomorphic to $G$ branched in $n$ points, parameterized by algebraic varieties $S$. When $G$ is with trivial center we prove that the Hurwitz space $H_n^G(Y)$ is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary $G$ we prove that $H_n^G(Y)$ is a coarse moduli variety. For families of pointed Galois covers of $(Y,y_0)$ we prove that the Hurwitz space $H_n^G(Y,y_0)$ is a fine moduli variety, and construct explicitly the universal family, for arbitrary group $G$. We use classical tools of algebraic topology and of complex algebraic geometry.

v6: 42 pages, manuscript accepted for publication

References in corpus (2)

• Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds
• A criterion for extending morphisms from open subsets of smooth fibrations of algebraic varieties