We compute a closed formula for the class of the closure of the locus of curves in Mg that admit an abelian differential of signature κ = (k1, ..., kg−2).

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We show $\overline{\mathcal{M}}_{10, 10}$ and $\overline{\mathcal{F}}_{11,9}$ have Kodaira dimension zero. Our method relies on the construction of a number of curves via nodal Lefschetz pencils on blown-up $K3$ surfaces. The construction further yields that any effective divisor in $\overline{\mathcal{M}}_{g}$ with slope $<6+(12-\delta )/(g+1)$ must contain the locus of curves that are the normalization of a $\delta $-nodal curve lying on a $K3$ surface of genus $g+\delta $.

For g ≥ 2, j = 1, . . . , g and n ≥ g + j we exhibit infinitely many new rigid and extremal effective codimension j cycles in Mg,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 ≥ 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 Mg,n with negative ψi coefficients.Contents 15 5. Rigidity and extremality of higher codimension cycles 19 References 22

We show M 10,10 and F 11,9 have Kodaira dimension zero. Our method utilised the construction of a number of curves via nodal Lefschetz pencils on blown-up K3 surfaces which further yield that any effective divisor in Mg with slope < 6 + (12 − δ)/(g + 1) must contain the locus of curves that are the normalization of a δ-nodal curve lying on a K3 surface of genus g + δ.