On automorphisms of Hilbert squares of smooth hypersurfaces

TL;DR

This paper proves that for smooth projective hypersurfaces of dimension at least three, all automorphisms of their Hilbert squares originate from automorphisms of the hypersurfaces themselves.

Contribution

It establishes a correspondence between automorphisms of the Hilbert square and automorphisms of the original hypersurface, extending understanding of automorphism groups.

Findings

Automorphisms of $X^{[2]}$ are induced by automorphisms of $X$

The result applies to hypersurfaces of dimension at least three

Provides insight into the structure of automorphism groups of Hilbert squares

Abstract

Let $X$ be a smooth projective hypersurface of dimension at least three. We show that every automorphism of the Hilbert square $X^{[2]}$ of $X$ is induced by some automorphism of $X$.