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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

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