A very special EPW sextic and two IHS fourfolds

Abstract

We show that the Hilbert scheme of two points on the Vinberg $K3$ surface has a two-to-one map onto a very symmetric EPW sextic $Y$ in ${\mathbb{P}}^5$. The fourfold $Y$ is singular along $60$ planes, $20$ of which form a complete family of incident planes. This solves a problem of Morin and O’Grady and establishes that $20$ is the maximal cardinality of such a family of planes. Next, we show that this Hilbert scheme is birationally isomorphic to the Kummer-type IHS fourfold $X_0$ constructed by Donten-Bury and Wiśniewski [On 81 symplectic resolutions of a 4–dimensional quotient by a group of order $32$, preprint (2014)]. We find that $X_0$ is also related to the Debarre–Varley abelian fourfold.