Fano congruences of index 3 and alternating 3-forms

De Poi, Pietro; Faenzi, Daniele; Mezzetti, Emilia; Ranestad, Kristian

Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 2099-2165.

Nous étudions des congruences de droites $X_{\omega }$ définies par une $3$-forme alternée suffisamment générale en $n+1$ variables. Celles-ci sont des variétés de Fano d’indice $3$ et dimension $n-1$. La classe de ces congruences contient la $5$-variété homogène sous ${\mathrm{G}}_2$ dans ${\mathbb{P}}^{13}$ pour $n=6$ et la variété des réductions d’une projection générique de ${\mathbb{P}}^2\times {\mathbb{P}}^2$ dans ${\mathbb{P}}^7$ pour $n=7$.

Nous montrons que le degré de $X_{\omega }$ est le $n$-ième nombre de Fine. Nous étudions le schéma de Hilbert de ces congruences et montrons que le choix de $\omega$ correspond birationnellement au choix de $X_{\omega }$ sauf si $n=5$.

Le lieu fondamental de ces congruences est étudié aussi bien que son lieu singulier  : la classe de ces variétés inclut la cubique de Coble pour $n=8$ et la variété de Peskine pour $n=9$.

La congruence résiduelle $Y$ de $X_{\omega }$ par rapport à une congruence linéaire générique contenant $X_{\omega }$ est analysée à travers les quadriques qui contiennent l’espace linéaire engendré par $X_{\omega }$. Nous montrons que $Y$ est Cohen–Macaulay mais pas Gorenstein en codimension $4$. Nous examinons le lieu fondamental $G$ de $Y$, duquel nous déterminons les singularités et les composantes irréductibles.

We study congruences of lines $X_{\omega }$ defined by a sufficiently general choice of an alternating 3-form $\omega$ 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 $X_{\omega }$ as the $n$-th Fine number and study the Hilbert scheme of these congruences proving that the choice of $\omega$ bijectively corresponds to $X_{\omega }$ 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 $X_{\omega }$ with respect to a general linear congruence containing $X_{\omega }$ is analysed in terms of the quadrics containing the linear span of $X_{\omega }$. 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.

Classification : 14M15, 14J45, 14J60, 14M06, 14M05
Keywords: Fano varieties; congruences of lines; trivectors; alternating 3-forms; Cohen–Macaulay varieties; linkage; linear congruences; Coble variety; Peskine variety; variety of reductions; secant lines; fundamental loci.
Mot clés : variétés de Fano ; congruences de droites ; trivecteurs ; 3-formes alternées ; variétés de Cohen-Macaulay ; liaison ; congruences linéaires ; variété de Coble ; variété de Peskine ; variétés de réduction ; droites sécantes ; lieu fondamental.

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