A special Debarre-Voisin fourfold

Abstract

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 ${\sigma }_0\in {\bigwedge }^3{V}_{10}^{\vee }$ that is $\mathbf{G}$-invariant. Following the construction of Debarre-Voisin, we obtain a smooth hyperkähler 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.

20 pages; published version, expanded with Section 6 on the moduli space

References in corpus (1)

• Some divisors in the moduli space of Debarre-Voisin varieties