Journal of the European Mathematical Society

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Volume 19, Issue 3, 2017, pp. 603–658
DOI: 10.4171/JEMS/677

Singularities of the moduli space of level curves

Alessandro Chiodo[1] and Gavril Farkas[2]

(1) Institut de Mathématiques de Jussieu, UMR 7586, Université Pierre et Marie Curie, 4, Place Jussieu, 75252, Paris CEDEX 05, France
(2) Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Raum 1.401, 10099, Berlin, Germany

We describe the singular locus of the compacti cation of the moduli space $\mathcal R_{g, \ell}$ of curves of genus $g$ paired with an $\ell$-torsion point in their Jacobian. Generalising previous work for $\ell ≤  2$, we also describe the sublocus of noncanonical singularities for any positive integer $\ell$. For $g$ ≥ 4 and $\ell$ = 3, 4, 6, this allows us to provide a lifting result on pluricanonical forms playing an essential role in the computation of the Kodaira dimension of $\mathcal R_{g, \ell}$: for those values of $\ell$, every pluricanonical form on the smooth locus of the moduli space extends to a desingularisation of the compacti ed moduli space.

Keywords: Moduli of curves

Chiodo Alessandro, Farkas Gavril: Singularities of the moduli space of level curves. J. Eur. Math. Soc. 19 (2017), 603-658. doi: 10.4171/JEMS/677