# Automorphisms of Hilbert schemes of points on surfaces

## TL;DR

This paper proves that automorphisms of Hilbert schemes of points on most surfaces are induced by surface automorphisms, with specific exceptions for product surfaces and the case of two points.

## Contribution

It establishes conditions under which automorphisms of Hilbert schemes are natural, extending previous understanding and identifying unique non-natural automorphisms in special cases.

## Key findings

- Automorphisms of Hilbert schemes on weak Fano or general type surfaces are mostly natural.
- Unique non-natural automorphism exists for product of curves when n=2.
- Automorphisms of Hilbert schemes of 2 points on projective space are all natural.

## Abstract

We show that every automorphism of the Hilbert scheme of $n$ points on a weak Fano or general type surface is natural, i.e. induced by an automorphism of the surface, unless the surface is a product of curves and $n=2$. In the exceptional case there exists a unique non-natural automorphism. More generally, we prove that any isomorphism between Hilbert schemes of points on smooth projective surfaces, where one of the surfaces is weak Fano or of general type and not equal to the product of curves, is natural. We also show that every automorphism of the Hilbert scheme of $2$ points on $\mathbb{P}^n$ is natural.