Original version
Transactions of the American Mathematical Society. 2020, 373 (9), 6139-6156, DOI: https://doi.org/10.1090/tran/8106
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 Pn is natural.