Moduli of G-covers of curves: geometry and singularities

Abstract

We analyze the singular locus and the locus of non-canonical singularities of the moduli space R g,G of curves with a G-cover for any finite group G. We show that non-canonical singularities are of two types: T -curves, that is singularities lifted from the moduli space Mg of stable curves, and J-curves, that is new singularities entirely characterized by the dual graph of the cover. Finally, we prove that in the case G = S 3 , the J-locus is empty, which is the first fundamental step in evaluating the Kodaira dimension of R g,S 3 .Résumé. -Nous analysons le lieu singulier et le lieu des singularités noncanoniques de l'espace de modules R g,G des courbes avec un G-recouvrement où G est un groupe fini. Nous montrons que les singularités non canoniques sont de deux types: T -courbes, c'est-à-dire des singularités relevées de l'espace de modules Mg des courbes stables, et J-courbes, c'est-à-dire des singularités nouvelles caractérisées entièrement par le graphe dual du recouvrement. Enfin, nous prouvons que dans le cas G = S 3 , le lieu J est vide, une première étage très importante dans l'évaluation de la dimension de Kodaira de R g,S 3

On the weight zero compactly supported cohomology of

“…Implementations in SageMath are available in the package [25]. Closely related, the notion of a graph G-cover associated to a admissible G -cover was developed by Galeotti [28, 29] – see especially [29, §3.1] – for the purpose of studying the birational geometry, and singularities, of (coarse spaces of) moduli spaces of genus g curves with a principal G -bundle. Our definition is a version of these, undertaken in a case when it becomes possible to explicitly determine the combinatorics of the connected strata of the boundary.…”

Abelian tropical covers

“…This idea seems to have appeared independently in other works as well: Chiodo and Farkas [ CF17 ] study the boundary of the moduli space of level curves, which is equivalent to a component of the moduli space of -admissible covers for a cyclic group , and look at cyclic covers of an arbitrary graph. Their work has been extended to an arbitrary finite group by Galeotti in [ Gal19a, Gal19b ]. Finally, in [ SvZ20 ], Schmitt and van Zelm apply a graph-theoretic approach to the boundary of the moduli space of -admissible covers (for an arbitrary finite group ) to study their pushforward classes in the tautological ring of .…”