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Line congruences of low degree. (English) Zbl 0632.14038
Géométrie algébrique et applications, C. R. 2ieme Conf. int., La Rabida/Espagne 1984, II: Singularités et géométrie complexe, Trav. Cours 23, 141-154 (1987).
[For the entire collection see Zbl 0614.00007.]
A line congruence is a surface in the Grassmann variety, \(G=Grass\) (1,3), of lines in projective 3-space. The bidegree of a line congruence S is \((d,d')\), where d is the number of lines of S passing through a (general) point of \({\mathbb{P}}^ 3\), and \(d'\) is the number of lines in a (general) plane. The authors classify all line congruences with degree \(d+d'\leq 5\), by describing the corresponding Hilbert schemes.
A. Papantonopoulou [Proc. Am. Math. Soc. 89, 583-586 (1983; Zbl 0572.14027) and 95, 533-536 (1985; Zbl 0626.14036)] gave a list of possible types of congruences of degree \(\leq 8\). She gave no proof of existence, however, and in the paper under review it is shown that the one of bidegree (2,2), isomorphic to \(F_ 2\), must be excluded.
Reviewer: R.Piene

14M15 Grassmannians, Schubert varieties, flag manifolds
14C05 Parametrization (Chow and Hilbert schemes)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)