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.