We describe the singular locus of the compactication of the moduli space of curves of genus paired with an -torsion point in their Jacobian. Generalising previous work for \( \ell ≤ 2 \), we also describe the sublocus of noncanonical singularities for any positive integer . For ≥ 4 and = 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 : for those values of , every pluricanonical form on the smooth locus of the moduli space extends to a desingularisation of the compactied moduli space.
Cite this article
Alessandro Chiodo, Gavril Farkas, Singularities of the moduli space of level curves. J. Eur. Math. Soc. 19 (2017), no. 3, pp. 603–658DOI 10.4171/JEMS/677