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

## TL;DR

This paper constructs new rigid and extremal effective cycles in the moduli space of curves using strata of differentials, and computes divisor classes revealing examples with negative coefficients.

## Contribution

It introduces infinitely many new rigid cycles in ar{\u03bc}_{g,n} from differential strata and computes their classes, including the first known effective divisors with negative bfi coefficients.

## Key findings

- Constructed infinitely many rigid cycles from differential strata.
- Computed divisor classes with negative bfi coefficients.
- Extended results to k-differentials for irreducible strata.

## Abstract

For $g\geq2$, $j=1,\dots,g$ and $n\geq g+j$ we exhibit infinitely many new rigid and extremal effective codimension $j$ cycles in $\overline{\mathcal{M}}_{g,n}$ from the strata of quadratic differentials and projections of these strata under forgetful morphisms and 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.