Classification of smooth congruences with a fundamental curve

Abstract

A congruence of lines is a (n−1)-dimensional family of lines in Pn (over C), i.e. a variety Y of dimension (and hence of codimension) n − 1 in the Grassmannian Gr(1, Pn). A fundamental curve for Y is a curve C Pn which meets all the lines of Y . In this paper the authors classify all smooth congruences with fundamental curve C generalizing a paper by E. Arrondo and M. Gross [Manuscr. 79, No. 3-4, 283-298 (1993; Zbl 0803.14019)], where the case n = 3 was treated. An explicit construction for all possible congruences that they found is also given.