# Geometry of the moduli space of $n$-pointed K3 surfaces of genus 11

## TL;DR

This paper investigates the geometric properties of the moduli space of genus 11 K3 surfaces with marked points, establishing conditions for unirationality, uniruledness, and non-negative Kodaira dimension.

## Contribution

It proves the unirationality for up to 6 marked points, uniruledness for up to 7, and non-negative Kodaira dimension for 9 marked points, resolving longstanding questions.

## Key findings

- Unirational for n ≤ 6
- Uniruled for n ≤ 7
- Non-negative Kodaira dimension at n=9

## Abstract

We prove that the moduli space of polarized $K3$ surfaces of genus eleven with $n$ marked points is unirational when $n\leq 6$ and uniruled when $n\leq7$. As a consequence, we settle a long standing but not proved assertion about the unirationality of $\cal{M}_{11,n}$ for $n\leq6$. We also prove that the moduli space of polarized $K3$ surfaces of genus eleven with $9$ marked points has non-negative Kodaira dimension.