Counting Rational Curves on K3 Surfaces With Finite Group Actions

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