$k$-differentials on curves and rigid cycles in moduli space

Mullane, Scott

Documenta mathematica, Tome 26 (2021), pp. 1817-1850.

Résumé

For $g\geq 2, j=1,\dots,g$ and $n\geq g+j$ we exhibit infinitely many new rigid and extremal effective codimension $j$ cycles in $\overline{\mathcal{M}}_{g,n} $, the Deligne-Mumford compactification of the moduli of $n$-pointed curves of genus $g$. The extremal cycles constructed correspond to the strata of quadratic differentials and projections of these strata under forgetful morphisms. We further show the same holds for $k$-differentials with $k\geq 3$ if the strata are irreducible. We compute the class of the divisors in the case of quadratic differentials which contain the first known examples of effective divisors on $\overline{\mathcal{M}}_{g,n}$ with negative $\psi_i$ coefficients.

Détail

Classification : 14C20, 14C25, 14E30
Keywords: moduli of curves, divisors, rigid cycles, Teichmüller dynamics