Automorphisms of the Hilbert schemes of n points of a rational surface and the anticanonical Iitaka dimension (Q776219)

In this paper, the author studies the relationship between the automorphism group of a rational surface and that of its Hilbert scheme of points. For a divisor \(D\) on a smooth projective variety \(X\), let \(\kappa(X, D)\) be the Iitaka dimension of \(D\) and \(K_X\) be the canonical divisor of \(X\). The Hilbert scheme \(S^{[n]}\) of \(n\) points on a smooth projective surface \(S\) is known to be smooth projective of dimension \(2n\). An automorphism of \(S^{[n]}\) is defined to be natural if it commutes with the Hilbert-Chow morphism \(S^{[n]} \to S^{(n)}\) where \(S^{(n)}\) is the \(n\)-th symmetric product of \(S\). Fix a rational surface \(S\) with \(\kappa(S, -K_S) \ge 1\). The author proves that if \(T\) is a smooth projective surface and \(S^{[n]} \cong T^{[n]}\) for some \(n \ge 2\), then \(S \cong T\). Moreover, if \(S \not \cong \mathbb P^1 \times \mathbb P^1\) and \(n \ge 2\), or if \(S = \mathbb P^1 \times \mathbb P^1\) and \(n \ge 3\), then all the automorphisms of \(S^{[n]}\) are natural (hence, \(\mathrm{Aut}(S^{[n]}) \cong\mathrm{Aut}(S)\)). The remaining case \((\mathbb P^1 \times \mathbb P^1)^{[2]}\) does have a non-natural automorphism which leaves the exceptional divisor of the Hilbert-Chow morphism globally invariant. The main idea in the proofs is to use the anti-canonical divisors and the properties of the Hilbert-Chow morphism.