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 → Y × S → S of Galois covers of Y with Galois group G branched in n points, parameterized by algebraic varieties S. When G is with trivial center we prove that the Hurwitz space H G n (Y ) is a fine moduli variety for this category and construct explicitly the universal family. For arbitrary G we prove that H G n (Y ) is a coarse moduli variety. For families of pointed Galois covers of (Y, y 0 ) we prove that the Hurwitz space H G n (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.