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 $g+\delta$.

12 pages. Minor corrections, to appear in IMRN