Automorphisms of Hilbert Schemes of Points on Abelian Surfaces

Belmans, Oberdieck, and Rennemo asked whether natural automorphisms of Hilbert schemes of points on surfaces can be characterized by the fact that they preserve the exceptional divisor of non-reduced subschemes. Sasaki recently published examples, independently discovered by the author, of automorphisms on the Hilbert scheme of two points of certain abelian surfaces which preserve the exceptional divisor but are nevertheless unnatural, giving a negative answer to the question. We construct additional examples for abelian surfaces of unnatural automorphisms which preserve the exceptional divisor of the Hilbert scheme of an arbitrary number of points. The underlying abelian surfaces in these examples have Picard rank at least 2, and hence are not generic. We prove the converse statement that all automorphisms are natural on the Hilbert scheme of two points for a principally polarized abelian surface of Picard rank 1. Additionally, we prove the same if the polarization has self-intersection a perfect square.