# Two moduli spaces of Calabi-Yau type

## TL;DR

This paper proves that certain moduli spaces of Calabi-Yau type have Kodaira dimension zero by constructing specific curves on blown-up K3 surfaces, revealing geometric properties of these spaces.

## Contribution

It introduces a novel method using nodal Lefschetz pencils on K3 surfaces to analyze the Kodaira dimension of specific moduli spaces.

## Key findings

- ar{ ext{M}}_{10,10} and ar{ ext{F}}_{11,9} have Kodaira dimension zero
- Effective divisors with slope below a certain threshold contain loci of nodal curve normalizations on K3 surfaces
- Construction of curves via Lefschetz pencils provides new tools for studying moduli space geometry

## 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 $g+\delta$.