Fano congruences of index $3$ and alternating $3$-forms

Abstract

We study congruences of lines $X_ω$ defined by a sufficiently general choice of an alternating 3-form $ω$ in $n+1$ dimensions, as Fano manifolds of index $3$ and dimension $n-1$. These congruences include the $\mathrm{G}_2$-variety for $n=6$ and the variety of reductions of projected $\mathbb{P}^2 \times \mathbb{P}^2$ for $n=7$. We compute the degree of $X_ω$ as the $n$-th Fine number and study the Hilbert scheme of these congruences proving that the choice of $ω$ bijectively corresponds to $X_ω$ except when $n=5$. The fundamental locus of the congruence is also studied together with its singular locus: these varieties include the Coble cubic for $n=8$ and the Peskine variety for $n=9$. The residual congruence $Y$ of $X_ω$ with respect to a general linear congruence containing $X_ω$ is analysed in terms of the quadrics containing the linear span of $X_ω$. We prove that $Y$ is Cohen-Macaulay but non-Gorenstein in codimension $4$. We also examine the fundamental locus $G$ of $Y$ of which we determine the singularities and the irreducible components.