The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Journal of the European Mathematical Society


Full-Text PDF (466 KB) | Metadata | Table of Contents | JEMS summary
Volume 19, Issue 3, 2017, pp. 603–658
DOI: 10.4171/JEMS/677

Published online: 2017-02-15

Singularities of the moduli space of level curves

Alessandro Chiodo[1] and Gavril Farkas[2]

(1) Université Pierre et Marie Curie, Paris, France
(2) Humboldt-Universität zu 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