# Fano congruences of index 3 and alternating 3-forms

Abstract : 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 $G_2$-variety for $n = 6$ and the variety of reductions of projected $\mathbb{P^2}$ x $\mathbb{P^2}$ 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.
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https://hal.archives-ouvertes.fr/hal-01661615
Contributor : Imb - Université de Bourgogne <>
Submitted on : Tuesday, December 12, 2017 - 8:35:44 AM
Last modification on : Tuesday, May 14, 2019 - 4:12:01 PM

### Citation

Pietro de Poi, Daniele Faenzi, Emilia Mezzetti, Kristian Ranestad. Fano congruences of index 3 and alternating 3-forms. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2017, 67 (5), pp.2099 - 2165. ⟨10.5802/aif.3131⟩. ⟨hal-01661615⟩

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