We study congruences of lines defined by a sufficiently general choice of an alternating 3-form in dimensions, as Fano manifolds of index and dimension . These congruences include the -variety for and the variety of reductions of projected for .
We compute the degree of as the -th Fine number and study the Hilbert scheme of these congruences proving that the choice of bijectively corresponds to except when . The fundamental locus of the congruence is also studied together with its singular locus: these varieties include the Coble cubic for and the Peskine variety for .
The residual congruence of with respect to a general linear congruence containing is analysed in terms of the quadrics containing the linear span of . We prove that is Cohen–Macaulay but non-Gorenstein in codimension . We also examine the fundamental locus of of which we determine the singularities and the irreducible components.
Nous étudions des congruences de droites définies par une -forme alternée suffisamment générale en variables. Celles-ci sont des variétés de Fano d’indice et dimension . La classe de ces congruences contient la -variété homogène sous dans pour et la variété des réductions d’une projection générique de dans pour .
Nous montrons que le degré de est le -ième nombre de Fine. Nous étudions le schéma de Hilbert de ces congruences et montrons que le choix de correspond birationnellement au choix de sauf si .
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 et la variété de Peskine pour .
La congruence résiduelle de par rapport à une congruence linéaire générique contenant est analysée à travers les quadriques qui contiennent l’espace linéaire engendré par . Nous montrons que est Cohen–Macaulay mais pas Gorenstein en codimension . Nous examinons le lieu fondamental de , duquel nous déterminons les singularités et les composantes irréductibles.
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Accepted:
Published online:
DOI: 10.5802/aif.3131
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.
@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}, pages = {2099--2165}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {67}, number = {5}, year = {2017}, doi = {10.5802/aif.3131}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.3131/} }
TY - JOUR AU - De Poi, Pietro AU - Faenzi, Daniele AU - Mezzetti, Emilia AU - Ranestad, Kristian TI - Fano congruences of index 3 and alternating 3-forms JO - Annales de l'Institut Fourier PY - 2017 SP - 2099 EP - 2165 VL - 67 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3131/ DO - 10.5802/aif.3131 LA - en ID - AIF_2017__67_5_2099_0 ER -
%0 Journal Article %A De Poi, Pietro %A Faenzi, Daniele %A Mezzetti, Emilia %A Ranestad, Kristian %T Fano congruences of index 3 and alternating 3-forms %J Annales de l'Institut Fourier %D 2017 %P 2099-2165 %V 67 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3131/ %R 10.5802/aif.3131 %G en %F 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/articles/10.5802/aif.3131/
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