On smooth surfaces in Gr\((1,\mathbb{P}^ 3)\) with a fundamental curve (Q1313511)

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 \textit{A. Verra}'s paper [ Manuscr. Math. 62, No. 4, 417-435 (1988; Zbl 0673.14026)].