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 2:1 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 in [DW]. We find that $X_0$ is also related to the Debarre-Varley abelian fourfold.