A very special EPW sextic and two IHS fourfolds

TL;DR

This paper explores the geometric properties of a special EPW sextic and related fourfolds, revealing maximal configurations of incident planes and connections to Hilbert schemes and abelian varieties.

Contribution

It demonstrates a 2:1 map from the Hilbert scheme of two points on the Vinberg K3 surface to a symmetric EPW sextic and establishes birational relations with IHS fourfolds.

Findings

Maximal family of 20 incident planes on the EPW sextic.

Hilbert scheme of two points on Vinberg K3 maps onto the EPW sextic.

Connections between the IHS fourfold and the Debarre-Varley abelian fourfold.

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.