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Classification of smooth congruences of low degree. (English) Zbl 0649.14027

We give a complete classification of smooth congruences - i.e. surfaces in the Grassmann variety of lines in \({\mathbb{P}}\) \(3_{{\mathbb{C}}}\) identified with a smooth quadric in \({\mathbb{P}}^ 5 \)- of degree \(at\quad most\quad 8,\) by studying which surfaces of \({\mathbb{P}}^ 5 \)can 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.
Reviewer: E.Arrondo

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14N05 Projective techniques in algebraic geometry
14J10 Families, moduli, classification: algebraic theory
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