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

TL;DR

This paper characterizes the geometric structure of Hilbert squares of generic K3 surfaces of degree 2t, showing they are double EPW sextics under certain conditions, thus linking K3 geometry with EPW sextics.

Contribution

It establishes a geometric description of Hilbert squares of generic K3 surfaces as double EPW sextics when specific ample divisors exist.

Findings

Hilbert squares of generic K3 surfaces of degree 2t are double EPW sextics for t > 2.

Existence of an ample divisor with q_X(D)=2 characterizes this geometric structure.

Provides a geometric classification linking K3 surfaces and EPW sextics.

Abstract

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.