Fano congruences of index 3 and alternating 3-forms

Abstract

We study congruences of lines Xω defined by a sufficiently general choice of an alternating 3-form ω in n dimensions, as Fano manifolds of index 3 and dimension n-1. These congruences include the G2-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.