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We present a sufficient condition for the punctural Hilbert scheme of lengthtwo of a K3 surface with finite automorphism group to have automorphism groupof infinite order in geometric terms (Theorem 2.1). We then give concreteexamples (Theorem 1.2). We also discuss about Mori dream space (MDS) structuresunder an extermal crepant resolution (Theorems 1.2, 5.2, 4.1) from theviewpoint of automorphisms. These affirmatively answer a question of DoctorMalte Wandel.

A question of Doctor Malte Wandel on automorphisms of the punctural Hilbert schemes of K3 surfaces