• Corpus ID: 54017607

Singularities of moduli of curves with a universal root

@article{Galeotti2015SingularitiesOM,
  title={Singularities of moduli of curves with a universal root},
  author={Mattia Galeotti},
  journal={arXiv: Algebraic Geometry},
  year={2015}
}

• Mattia Galeotti
• Published 2 April 2015
• Mathematics
• arXiv: Algebraic Geometry

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 $\ell\leq 6$ and $\ell\neq 5$ pluricanonical forms extend over any desingularization. This allows to compute the Kodaira dimension without desingularizing, as done by Farkas and Ludwig for $\ell=2$, and by Chiodo, Eisenbud, Farkas and Schreyer for $\ell=3$. 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 $L^{\otimes\ell}\cong\omega_C^{\otimes k}$. New loci of canonical and non-canonical singularities appear for any $k\not\in\ell\mathbb{Z}$ and $\ell>2$, we provide a set of combinatorial tools allowing us to completely describe the singular locus in terms of dual graph. We characterize the locus of non-canonical singularities, and for small values of $\ell$ we give an explicit description.