Automorphisms of Hilbert schemes of points on surfaces

Belmans, Pieter; Oberdieck, Georg; Rennemo, Jørgen

doi:10.1090/tran/8106

Automorphisms of Hilbert schemes of points on surfaces
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by Pieter Belmans, Georg Oberdieck and Jørgen Vold Rennemo PDF

Trans. Amer. Math. Soc. 373 (2020), 6139-6156 Request permission

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 nonnatural 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.

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