Arrondo, Enrique; Sols, Ignacio Classification of smooth congruences of low degree. (English) Zbl 0649.14027 J. Reine Angew. Math. 393, 199-219 (1989). 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 Cited in 2 ReviewsCited in 14 Documents MSC: 14M15 Grassmannians, Schubert varieties, flag manifolds 14N05 Projective techniques in algebraic geometry 14J10 Families, moduli, classification: algebraic theory Keywords:smooth congruences; surfaces in the Grassmann variety of lines; cohomology; postulation; linear normality; Hilbert scheme PDFBibTeX XMLCite \textit{E. Arrondo} and \textit{I. Sols}, J. Reine Angew. Math. 393, 199--219 (1989; Zbl 0649.14027) Full Text: Crelle EuDML