In a recent paper Chiodo and Farkas described the singular locus and the locus of non-canonical singularities of the moduli space of level curves. In this work we generalize their results to the moduli space Rg,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 = S3, the J-locus is empty, which is the first fundamental step in evaluating the Kodaira dimension of Rg,S 3 .

No 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

In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack [Formula: see text] of abstract Kummer varieties and the second one is the stack [Formula: see text] of embedded Kummer varieties. We will prove that [Formula: see text] is a Deligne-Mumford stack and its coarse moduli space is isomorphic to [Formula: see text], the coarse moduli space of principally polarized abelian varieties of dimension [Formula: see text]. On the other hand, we give a modular family [Formula: see text] of embedded Kummer varieties embedded in [Formula: see text], meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber. As a consequence, we construct the coarse moduli space [Formula: see text] of embedded Kummer surfaces and prove that it is obtained from [Formula: see text] by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: [Formula: see text] could be obtained from [Formula: see text] via a contraction for all [Formula: see text].

In a series of recent papers, Chiodo, Farkas and Ludwig carry out a deep analysis of the singular locus of the moduli space of stable (twisted) curves with an ℓ-torsion line bundle. They show that for ℓ ≤ 6 and ℓ = 5 pluricanonical forms extend over any desingularization. This opens the way to a computation of the Kodaira dimension without desingularizing, as done by Farkas and Ludwig for ℓ = 2, and by Chiodo, Eisenbud, Farkas and Schreyer for ℓ = 3. Here we treat roots of line bundles on the universal curve systematically: we consider the moduli space of curves C with a line bundle L such that L ⊗ℓ ∼ = ω ⊗k C . New loci of canonical and non-canonical singularities appear for any k ∈ ℓZ and ℓ > 2, we provide a set of combinatorial tools allowing us to completely describe the singular locus in terms of dual graphs. We characterize the locus of non-canonical singularities, and for small values of ℓ we give an explicit description.