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

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 $F$.