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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 \(H^G_n(Y)\) is a fine moduli variety for this moduli problem 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.

2010 Mathematics Subject Classification:

14H30, 14H10, 14D22

Keywords

Galois cover of a curve, family of covers, Hurwitz space, moduli space

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Author Details

Vassil Kanev

Dipartimento di Matematica e Informatica
Università di Palermo
Via Archirafi, 34
90123 Palermo, Italy
e-mail: vassil.kanev@unipa.it


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