EPW sextics and Hilbert squares of K3 surfaces

Abstract

We prove that the Hilbert square $S^{[2]}$ of a very general primitively polarized K3 surface S of degree $d(n)=2(4n^2+8n+5)$, $n\ge 1$ 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 $r^2+2$, $r\ge 1$ there is an antisymplectic involution. We explicitly give this involution on $S^{[2]}$ in term of the corresponding EPW polarization on it.

9 pages, typos corrected, introduction rewritten, abstract updated

References in corpus (1)

• Mori cones of holomorphic symplectic varieties of K3 type