Geometric description of $\langle 2 \rangle$-polarised Hilbert squares of generic K3 surfaces

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