# Fano Congruences

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Pietro De Poi

created by risa on 09 Mar 2018

15 mar 2018
-- 14:30

Aula 211, Dip.Matematica, Università "Roma Tre", Roma

**Abstract.**

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.