Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve
Serdica Mathematical Journal, Tome 50 (2024) no. 1, pp. 47-102
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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.
Keywords:
Galois cover of a curve, family of covers, Hurwitz space, moduli space, 14H30, 14H10, 14D22
Kanev, Vassil. Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve. Serdica Mathematical Journal, Tome 50 (2024) no. 1, pp. 47-102. http://geodesic.mathdoc.fr/item/SMJ2_2024_50_1_a2/
@article{SMJ2_2024_50_1_a2,
author = {Kanev, Vassil},
title = {Hurwitz moduli varieties parameterizing {Galois} covers of an algebraic curve},
journal = {Serdica Mathematical Journal},
pages = {47--102},
year = {2024},
volume = {50},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SMJ2_2024_50_1_a2/}
}