Surfaces of bidegree \((3,n)\) in \(\text{Gr}(1,\mathbb{P}^ 3)\) (Q1319297)

A congruence is a 2-dimensional family of lines in \(\mathbb{P}^ 3\), i.e., a surface \(Y\) in \(Gr(1,\mathbb{P}^ 3)\). Its bidegree is \((a,b)\), where \(a\) is the number of its lines passing through a general point, and \(b\) the number of its lines contained in a general plane. The purpose of the present paper is to classify all congruences of bidegree \((3,n)\). It follows from a previous result by the author that \(n \leq 9\). He now proves that if the congruence is smooth, then in fact \(n \leq 7\). For \(n \leq 5\), the classification is due to \textit{E. Arrondo} and \textit{I. Sols} [J. Reine Angew. Math. 393, 199-219 (1989; Zbl 0649.14027)], for \(n = 6\) most of it was done by \textit{A. Verra}, [Manuscr. Math. 62, No. 4, 417-435 (1988; Zbl 0673.14026)], whereas for \(n = 7\) the results are new.