Singularities of the moduli space of level curves
Abstract
We describe the singular locus of the compactification of the moduli space
R
g,l
of curves of genus
g
paired with an
l
-torsion point in their Jacobian. Generalising previous work for
l≤2
, we also describe the sublocus of noncanonical singularities for any positive integer
l
. For
g≥4
and
l=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
R
g,l
: for those values of
l
, every pluricanonical form on the smooth locus of the moduli space extends to a desingularisation of the compactified moduli space.