A generic K3 surface of degree 2t is a general complex projective K3 surface S 2t whose Picard group is generated by the class of an ample divisor H ∈ Div(S 2t ) such that H 2 = 2t with respect to the intersection form. We show that if X is the Hilbert square of a generic K3 surface of degree 2t with t = 2 which admits an ample divisor D ∈ Div(X) with q X (D) = 2, where q X is the Beauville-Bogomolov-Fujiki form, then X is a double EPW sextic.
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