Rational curves on K3 surfaces of small genus

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 $4$.