Two moduli spaces of Calabi-Yau type

Abstract

We show $\overline{\mathcal{M}}_{10,10}$ and $\mathcal{F}_{11,9}$ 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 $g+\delta$.