EPW sextics and Hilbert squares of K3 surfaces

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.

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