A special Debarre-Voisin fourfold

TL;DR

This paper investigates a highly symmetric hyperk"ahler fourfold derived from a special G-invariant trivector in a 10-dimensional representation of PSL(2,11), revealing its geometric properties and symmetries.

Contribution

It constructs and analyzes a new special Debarre-Voisin fourfold with notable symmetries and studies its associated Peskine variety with singularities.

Findings

The fourfold is smooth and hyperk"ahler with many symmetries.

The associated Peskine variety has 55 isolated singular points.

The geometry of the fourfold is better understood through the Peskine variety.

Abstract

Let $\mathbf G$ be the finite simple group $\mathrm{PSL}(2,\mathbf F_{11})$. It has an irreducible representation $V_{10}$ of dimension 10. In this note, we study a special trivector $\sigma\in \bigwedge^3V_{10}^\vee$ which is $\mathbf G$-invariant. Following the construction of Debarre-Voisin, we obtain a smooth hyperk\"ahler fourfold $X_6^\sigma\subset\mathrm{Gr}(6,V_{10})$ with many symmetries. We will also look at the associated Peskine variety $X_1^\sigma\subset \mathbf P(V_{10})$, which is highly symmetric as well and admits 55 isolated singular points. It will help us to understand better the geometry of the special Debarre-Voisin fourfold $X_6^\sigma$.