Fano congruences of index 3 and alternating 3-forms

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

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

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

Abstract
Résumé

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 $G_{2}$ -variety for $n = 6$ and the variety of reductions of projected $ℙ^{2} \times ℙ^{2}$ 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.

Nous étudions des congruences de droites $X_{ω}$ 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 $G_{2}$ dans $ℙ^{13}$ pour $n = 6$ et la variété des réductions d’une projection générique de $ℙ^{2} \times ℙ^{2}$ dans $ℙ^{7}$ pour $n = 7$ .

Nous montrons que le degré de $X_{ω}$ est le $n$ -ième nombre de Fine. Nous étudions le schéma de Hilbert de ces congruences et montrons que le choix de $ω$ correspond birationnellement au choix de $X_{ω}$ 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 $X_{ω}$ par rapport à une congruence linéaire générique contenant $X_{ω}$ est analysée à travers les quadriques qui contiennent l’espace linéaire engendré par $X_{ω}$ . 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.

Received : 2016-06-24
Revised : 2016-12-19
Accepted : 2017-01-24
Published online : 2017-11-17
DOI : https://doi.org/10.5802/aif.3131
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.

@article{AIF_2017__67_5_2099_0,
     author = {De Poi, Pietro and Faenzi, Daniele and Mezzetti, Emilia and Ranestad, Kristian},
     title = {Fano congruences of index 3 and alternating 3-forms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     pages = {2099-2165},
     doi = {10.5802/aif.3131},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_5_2099_0}
}

De Poi, Pietro; Faenzi, Daniele; Mezzetti, Emilia; Ranestad, Kristian. Fano congruences of index 3 and alternating 3-forms. Annales de l'Institut Fourier, Volume 67 (2017) no. 5, pp. 2099-2165. doi : 10.5802/aif.3131. http://www.numdam.org/item/AIF_2017__67_5_2099_0/

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