Fano congruences of index 3 and alternating 3-forms
[Congruences de droites de Fano d’indice 3 et 3-formes alternées]
Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 2099-2165.

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 13 pour n=6 et la variété des réductions d’une projection générique de 2 × 2 dans 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.

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 2 × 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.

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DOI : 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.
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.
De Poi, Pietro 1 ; Faenzi, Daniele 2 ; Mezzetti, Emilia 3 ; Ranestad, Kristian 4

1 Dipartimento di Scienze Matematiche, Informatiche e Fisiche Università degli Studi di Udine Via delle Scienze 206 Località Rizzi 33100 Udine (Italy)
2 Université de Bourgogne Institut de Mathématiques de Bourgogne UMR CNRS 5584 UFR Sciences et Techniques – Bâtiment Mirande – Bureau 310 9 Avenue Alain Savary BP 47870 21078 Dijon Cedex (France)
3 Dipartimento di Matematica e Geoscienze Sezione di Matematica e Informatica Università degli Studi di Trieste Via Valerio 12/1 34127 Trieste (Italy)
4 Department of Mathematics University of Oslo P.O. Box 1053 Blindern NO-0316 Oslo (Norway)
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De Poi, Pietro; Faenzi, Daniele; Mezzetti, Emilia; Ranestad, Kristian. Fano congruences of index 3 and alternating 3-forms. Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 2099-2165. doi : 10.5802/aif.3131. http://www.numdam.org/articles/10.5802/aif.3131/

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