Automorphisms of some \(K3\) surfaces and Beauville involutions of their Hilbert schemes (Q6161928)

The author determines the automorphism groups of certain \(K3\) surfaces \(X\) of Picard rank \(2\), and the automorphism groups of their Hilbert schemes \(X^{[2]}\) of length \(2\). The automorphisms of \(X\) are constructed by generalizing the Cayley-Oguiso's automorphism, and they induce natural automorphisms on \(X^{[2]}\). Moreover, on such \(K3\) surfaces \(X\), very ample divisors inducing isomorphisms with a quartic hypersurface define Beauville involutions of the Hilbert schemes \(X^{[2]}\). The author determines the relations between the natural automorphisms and the Beauville involutions on \(X^{[2]}\).