Fano congruences of index 3 and alternating 3-forms

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 × 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.