Counting rational curves on K3 surfaces with finite group actions

Sailun Zhan

Published 2019-07-07Version 1

G\"ottsche gave a formula for the dimension of the cohomology of Hilbert schemes of points on a smooth projective surface $S$. When $S$ admits an action by a finite group $G$, we describe the action of $G$ on the cohomology spaces. In the case that $S$ is a K3 surface, each element of $G$ gives a trace on $\sum_{n=0}\sum_{i=0}(-1)^{i}H^{i}(S^{[n]},\mathbb{C})q^{n}$. When $G$ acts faithfully and symplectically on $S$, the resulting generating function is of the form $q/f(q)$, where $f(q)$ is a cusp form. We relate the cohomology of Hilbert schemes of points to the cohomology of the compactified Jacobian of the tautological family of curves over an integral linear system on a K3 surface as $G$-representations. We give a sufficient condition for a $G$-orbit of curves with nodal singularities not to contribute to the representation.