Abstract
We present a sufficient condition for the punctural Hilbert scheme of length two of a K3 surface with finite automorphism group to have automorphism group of infinite order in geometric terms (Theorem 2.1) and supply it with concrete examples (Theorem 1.2). We also discuss Mori dream space structures under an extermal crepant resolution (Theorems 1.2, 4.1, 5.2) from the viewpoint of automorphisms. These affirmatively answer a question of Malte Wandel.
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