Abstract
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 $.
On the Connectedness of the Real Part of Moduli Spaces of Vector Bundles on Real Algebraic Surfaces