Two moduli spaces of Calabi-Yau type

Ignacio Barros, Scott Mullane

Published 2019-05-03Version 1

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