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On congruences of lines in the projective space. (Chapter 6 written in collaboration with M. Pedreira). (English) Zbl 0804.14016
The paper contains, in detail, the study of congruences of lines in \(\mathbb{P}^ 3\), that is of surfaces in the grassmannian \(G=Gr(1,3)\). To classify these surfaces is a classical problem; the interest in it is now mainly motivated by the analogy with the similar one for surfaces in \(\mathbb{P}^ 4\). Among the results obtained we note the following:
(a) A complete classification of smooth congruences up to degree \(d=9\) (result essentially known (Fano: \(d \leq 8)\) (Verra: \(d=9))\).
(b) It is a classical result that the Veronese surface is the only surface in \(\mathbb{P}^ 5\) which has a smooth projection in \(\mathbb{P}^ 4\) of the same degree. A similar problem is solved for surfaces \(F \subset G\), “generic projection” of a surface in \(Gr(1,4)\). (Is it possible to generalize in some way also Zak’s results?)
(c) A generalization to surfaces \(F \subset G\) of a result of Ellingsrud and Peskine for surfaces in \(\mathbb{P}^ 4\). The main part of it is that, except for a finite number of components, each component of the Hilbert scheme of smooth congruences consists of surfaces of general type.

MSC:
14J10 Families, moduli, classification: algebraic theory
14E05 Rational and birational maps
14M15 Grassmannians, Schubert varieties, flag manifolds
14J25 Special surfaces
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