#### Volume 21, issue 2 (2017)

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A very special EPW sextic and two IHS fourfolds

### Maria Donten-Bury, Bert van Geemen, Grzegorz Kapustka, Michał Kapustka and Jarosław A Wiśniewski

Geometry & Topology 21 (2017) 1179–1230
##### 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 ${ℙ}^{5}$. 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.

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##### Keywords
EPW sextics, IHS fourfolds, abelian varieties
##### Mathematical Subject Classification 2010
Primary: 14D06, 14J35, 14J70, 14K12, 14M07
Secondary: 14J50, 14J28