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On smooth surfaces in Gr\((1,\mathbb{P}^ 3)\) with a fundamental curve. (English) Zbl 0803.14019
Let \(Y\) be a smooth congruence of lines in complex projective 3-space. Assume that \(Y\) has a curve \(C\) of singular points. The authors give a complete classification of all such congruences. These are four classes which are – roughly speaking as follows:
(I) \(C\) is a line;
(II) \(Y\) is the set of bisecants of \(C\), with \(C\) a twisted cubic or an elliptic quartic;
(III) \(C\) is a scroll with \(C\) being a conic or a smooth plane cubic;
(IV) \(Y\) is a conic bundle over a smooth plane cubic.
In this classification there is one type of congruences that is missing in A. Verra’s paper [ Manuscr. Math. 62, No. 4, 417-435 (1988; Zbl 0673.14026)].
Reviewer: H.Havlicek (Wien)

14J25 Special surfaces
14M15 Grassmannians, Schubert varieties, flag manifolds
Full Text: DOI EuDML
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