Geometric description of 〈2〉-polarised Hilbert squares of generic K3 surfaces

Abstract

A generic K3 surface of degree 2t is a general complex projective K3 surface S2t whose Picard group is generated by the class of an ample divisor H∈Div(S2t) such that H2=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 qX(D)=2, where qX is the Beauville–Bogomolov–Fujiki form, then X is a double EPW sextic.