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

Automorphisms of Hilbert schemes of points on surfaces