Let \(S\) be a \(K3\) surface of Picard number \(2\), and denote by \(\mathrm{Hilb}^2(S)\) the Hilbert scheme of \(0\)-dimensional closed subschemes of length two on \(S\). In his previous paper [J. Alg. Geom. 23, No. 4, 775–795 (2014; Zbl 1304.14051)], the author proved that the group of automorphisms of \(S\) is of finite order. In the paper under review the author gives a sufficient condition in geometric terms for \(\mathrm{Hilb}^2(S)\) to have automorphism group of infinite order. He also finds a concrete example of \(K3\) surfaces \(S\) with \(|\mathrm{Aut}(\mathrm{Hilb}^2(S))|=\infty\). In his example, the Néron-Severi group of \(S\) is isomorphic to a certain even hyperbolic lattice of rank \(2\) and of discriminant \(17\).
As an interesting application for Mori dream space, the author shows that for a \(K3\) surface \(S\) in his example, the Hilbert-Chow morphism \(\mathrm{Hilb}^2(S)\to \mathrm{Sym}^2(S)\) is an extremal crepant resolution such that the source \(\mathrm{Hilb}^2(S)\) is not a Mori dream space but the target \(\mathrm{Sym}^2(S)\) is a Mori dream space.