Singularities of Generic Weingarten Congruences

The class of W-congruences is a central object of Projective Differential Geometry. Nevertheless, their singularities has not been extensively studied. In this paper we propose a new characterization of W-congruences that allow us to study their discriminant and umbilical points. We give examples of isolated stable umbilical points of type Aₘ, m∈\1,2,3,4\, and discuss some partial results concerning the generalizing of these results to any m∪\∞\.