##### Department of Mathematics,

University of California San Diego

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### Thesis Defense

## Patrick Girardet

#### UCSD

## Automorphisms of Hilbert Schemes of Points of Abelian Surfaces

##### Abstract:

Given an automorphism of a variety $X$, there is an induced ''natural'' automorphism on $X^{[n]}$, the Hilbert scheme of $n$ points of $X$. While unnatural automorphisms of $X^{[n]}$ are known to exist for certain varieties $X$ and integers $n$, all previously known examples could be shown to be unnatural because they do not preserve multiplicities. Belmans, Oberdieck, and Rennemo thus asked if an automorphism of a Hilbert scheme of points of a surface is natural if and only if it preserves the diagonal of non-reduced subschemes.

We give an answer in the negative for all $n\ge 2$ by constructing explicit counterexamples on certain abelian surfaces $X$. These surfaces are not generic, and hence we prove a partial converse statement that all automorphisms of the Hilbert scheme of two points on a very general abelian surface are natural for certain polarization types (including the principally polarized case).

Advisor: Dragos Oprea

### May 16, 2024

### 2:00 PM

APM 7321

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