On first order Congruences of Lines in ${\mathbb{P}}^4$ with irreducible fundamental Surface

Abstract

In this article we study congruences of lines in ${\mathbb{P}}^n$, and in particular of order one. After giving general results, we obtain a complete classification in the case of ${\mathbb{P}}^4$ in which the fundamental surface $F$ is in fact a variety-i.e. it is integral-and the congruence is the irreducible set of the trisecant lines of $F$.

18 pages, AMS-LaTeX; submitted