Automorphisms of Hilbert Schemes of Points on Abelian Surfaces

TL;DR

This paper investigates automorphisms of Hilbert schemes of points on abelian surfaces, providing examples of unnatural automorphisms and characterizing when automorphisms are necessarily natural based on the surface's polarization properties.

Contribution

It constructs new examples of unnatural automorphisms for Hilbert schemes on abelian surfaces with higher Picard rank and proves conditions under which all automorphisms are natural.

Findings

Examples of unnatural automorphisms for arbitrary number of points

Automorphisms are natural for Picard rank 1 surfaces with certain polarizations

Characterization of when automorphisms preserve the exceptional divisor

Abstract

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.