Classification of smooth congruences of low degree

Abstract

We give a complete classification of smooth congruences - i.e. surfaces in the Grassmann
variety of lines in P 3C identified with a smooth quadric in P5- of degree at most 8, by
studying which surfaces of P5can lie in a smooth quadric and proving their existence.
We present their ideal sheaf as a quotient of natural bundles in the Grassmannian,
what provides a perfect knowledge of its cohomology (for example postulation or linear
normality), as well as many information on the Hilbert scheme of these families, such
as dimension, smoothness, unirationality - and thus irreducibility - and in some cases
rationality.