Geodesic
Singularities of Moduli of Curves with a Universal Root
Documenta mathematica, Tome 22 (2017), pp. 1337-1373
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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≅ωC⊗k​. 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.
DOI : 10.4171/dm/599
Classification : 14H10, 14H20, 14H60
Mots-clés : moduli space of stable curves, torsion line bundles, canonical singularities 
@article{10_4171_dm_599,
     author = {Mattia Galeotti},
     title = {Singularities of {Moduli} of {Curves} with a {Universal} {Root}},
     journal = {Documenta mathematica},
     pages = {1337--1373},
     year = {2017},
     volume = {22},
     doi = {10.4171/dm/599},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/599/}
}
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Mattia Galeotti. Singularities of Moduli of Curves with a Universal Root. Documenta mathematica, Tome 22 (2017), pp. 1337-1373. doi: 10.4171/dm/599

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