Automorphisms of Hilbert schemes of points on surfaces

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.

20 pages; added reference to related work of Hayashi

References in corpus (3)

• Hochschild cohomology of noncommutative planes and quadrics
• A question of Doctor Malte Wandel on automorphisms of the punctural Hilbert schemes of K3 surfaces
• Strongly ambiguous Hilbert squares of projective K3 surfaces with Picard number one

Cited by in corpus (1)

• Derived categories of flips and cubic hypersurfaces