EPW sextics and Hilbert squares of K3 surfaces

TL;DR

The paper demonstrates that the Hilbert square of a very general primitively polarized K3 surface with specific degrees is birational to a double EPW sextic, confirming a conjecture about antisymplectic involutions for certain K3 surfaces.

Contribution

It establishes a birational equivalence between Hilbert squares of K3 surfaces and double EPW sextics for specific degrees, and explicitly constructs the associated antisymplectic involution.

Findings

Hilbert square of K3 surfaces with degree $d(n)$ is birational to a double EPW sextic.

Confirms O'Grady's conjecture for even $r$, regarding antisymplectic involutions.

Provides explicit description of the involution in terms of EPW polarization.

Abstract

We prove that the Hilbert square $S^{[2]}$ of a very general primitively polarized K3 surface S of degree $d(n) = 2(4n^2 + 8n + 5)$, $n \geq 1$ is birational to a double Eisenbud-Popescu-Walter sextic. Our result implies a positive answers, in the case when $r$ is even, to a conjecture of O'Grady: On the Hilbert square of a very general K3 surface of genus $r^2 + 2$, $r \geq 1$ there is an antisymplectic involution. We explicitly give this involution on $S^{[2]}$ in term of the corresponding EPW polarization on it.