Fano congruences
Pietro de Poi (Universita' di Udine)

We study congruences of lines of P^n (i.e. subvarieties of the Grassmannian of (co)dimension n-1) X defined by 3-forms, a class of congruences that are irreducible components of some reducible linear congruences, and their residual Y. We prove that X, and its fundamental locus F if n is odd, are Fano varieties of index 3 and that X is smooth; F is smooth as well if n<10. We study the Hilbert scheme of these congruences X, proving that the choice of the 3-form bijectively corresponds to X, except when n=5. Y is analysed in terms of the quadrics containing the linear span of X and we determine the singularities and the irreducible components of its fundamental locus. Joint work with Emilia Mezzetti, Daniele Faenzi and Kristian Ranestad.

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