Automorphisms of Hilbert schemes of points on a generic projective K3 surface

TL;DR

This paper investigates the automorphism groups of Hilbert schemes of points on generic projective K3 surfaces, revealing conditions under which these groups are trivial or generated by specific involutions, and demonstrating the existence of non-natural involutions for infinitely many degrees.

Contribution

It characterizes automorphism groups of Hilbert schemes on generic K3 surfaces and identifies conditions for non-trivial automorphisms, extending previous results to all n ≥ 2.

Findings

Automorphism group is either trivial or generated by a non-symplectic involution.

Numerical and divisorial conditions distinguish the two cases.

Existence of non-natural involutions for infinitely many degrees of S.

Abstract

We study automorphisms of the Hilbert scheme of $n$ points on a generic projective K3 surface $S$, for any $n \geq 2$. We show that the automorphism group of $S^{[n]}$ is either trivial or generated by a non-symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. As an application of these results we prove that, for any $n \geq 2$, there exist infinite values for the degree of $S$ such that $S^{[n]}$ admits a non-natural involution. This provides a generalization of results by Boissi\`ere--Cattaneo--Nieper-Wisskirchen--Sarti for $n=2$.