FANO CONGRUENCES OF INDEX 3 AND ALTERNATING 3-FORMS

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 G2-variety for n = 6 and the variety of reductions of projected P-2 x 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.