Singularities of Moduli of Curves with a Universal Root

  • Mattia Galeotti

    Institut de Mathematiques de Jussieu, UPMC, 4 Place Jussieu, 75005 Paris, France
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Abstract

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 \ell-torsion line bundle. They show that for 6\ell\leq 6 and 5\ell\ne 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 =2\ell=2, and by Chiodo, Eisenbud, Farkas and Schreyer for =3\ell=3. Here we treat roots of line bundles on the universal curve systematically: we consider the moduli space of curves CC with a line bundle LL such that LωCkL^{\otimes\ell}\cong \omega_C^{\otimes k}. New loci of canonical and non-canonical singularities appear for any k∉Zk\not\in\ell\Bbb Z and >2\ell>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 \ell we give an explicit description.

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Mattia Galeotti, Singularities of Moduli of Curves with a Universal Root. Doc. Math. 22 (2017), pp. 1337–1373

DOI 10.4171/DM/599