Rational curves on K3 surfaces of small genus

TL;DR

This paper investigates the monodromy groups of rational curves on primitively polarized K3 surfaces of small genus, specifically solving the cases for genus 3 and 4, extending known results for genus 2.

Contribution

It determines the monodromy groups of rational curves on K3 surfaces for genus 3 and 4, expanding understanding beyond the previously known genus 2 case.

Findings

Monodromy group for genus 3 is computed.

Monodromy group for genus 4 is computed.

Extends known results from genus 2 to higher small genera.

Abstract

Let $\mathfrak B_g$ denote the moduli space of primitively polarized $K3$ surfaces $(S,H)$ of genus $g$ over $\mathbb C$. It is well-known that $\mathfrak B_g$ is irreducible and that there are only finitely many rational curves in $|H|$ for any primitively polarized $K3$ surface $(S,H)$. So we can ask the question of finding the monodromy group of such curves. The case of $g=2$ essentially follows from the results of Harris \cite{Ha} to be the full symmetric group $S_{324}$, here we solve the case $g=3$ and $4$.