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Moduli of curves with principal and spin bundles : the singular locus via graph theory

Mardi 13 mars 11:00-12:00 - Mattia GALEOTTI - IMJ, Paris

Moduli of curves with principal and spin bundles : the singular locus via graph theory

Résumé : In a series of recent papers, Chiodo, Farkas and Ludwig carried out a
deep analysis of the singular locus of the moduli space of stable (twisted) curves
with an $\ell$-torsion line bundle. They showed that for $\ell\leq 6$ and $\ell\neq 5$
pluricanonical forms extend over any desingularization.
This opens the way to a computation of the Kodaira dimension without desingularizing,
as done by Farkas and Ludwig for $\ell=2$,
and by Chiodo, Eisenbud, Farkas and Schreyer
for $\ell=3$.
We can generalize this works in two directions.
At first 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 $L^\xx\ell\cong\omega_C^\xx k$.
New loci of canonical and non-canonical singularities appear
for any $k\not\in\ell\Z$ and $\ell>2$, we provide a set of combinatorial tools allowing us
to completely describe the singular locus in terms of dual graphs.
Furthermore, we treat moduli spaces of curves with a $G$-cover
where $G$ is any finite group. In particular for $G=S_3$ we approach
the evaluation of the Kodaira dimension of the moduli space, and list
the remaining obstacles to calculate it.

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