Monodromy of rational curves on K3 surfaces of low genus

Sailun Zhan

Published 2020-04-18Version 1

The monodromy group of the flexes or bitangents of a generic plane curve, the monodromy group of the lines on a generic degree $2n-3$ hypersurface in $\mathbb{P}^{n}$, and the monodromy group of the conics tangent to five generic conics are well-studied. In most cases, these monodromy groups are the full symmetric groups. We study a similar phenomenon on the rational curves in $|\mathcal{O}(1)|$ on a generic K3 surface. We prove that when the K3 surface has genus $g$, $1\leq g\leq 3$, the monodromy group is also the full symmetric group.