A special Debarre-Voisin fourfold

Consider the finite simple group $\mathbf{G}:=\mathrm{PSL}(2,\mathbf{F}_{11})$ of order 660, which has an irreducible representation $V_{10}$ of dimension 10. In this note, we study a special trivector ${\sigma_0}\in \bigwedge^3V_{10}^\vee$ that is $\mathbf{G}$-invariant. Following the construction of Debarre-Voisin, we obtain a smooth hyperk\"ahler fourfold $X_6^{\sigma_0}\subset\mathrm{Gr}(6,V_{10})$ with many symmetries. We will also look at the associated Peskine variety $X_1^{\sigma_0}\subset \mathbf{P}(V_{10})$, which is highly symmetric as well and admits 55 isolated singular points. It will help us to better understand the geometry of the special Debarre-Voisin fourfold $X_6^{\sigma_0}$. We also discuss an application of this example to the global geometry of the moduli space of Debarre-Voisin fourfolds.