Secants, bitangents, and their congruences

A congruence is a surface in the Grassmannian $\mathrm{Gr}(1,\mathbb{P}^3)$ of lines in projective $3$-space. To a space curve $C$, we associate the Chow hypersurface in $\mathrm{Gr}(1,\mathbb{P}^3)$ consisting of all lines which intersect $C$. We compute the singular locus of this hypersurface, which contains the congruence of all secants to $C$. A surface $S$ in $\mathbb{P}^3$ defines the Hurwitz hypersurface in $\mathrm{Gr}(1,\mathbb{P}^3)$ of all lines which are tangent to $S$. We show that its singular locus has two components for general enough $S$: the congruence of bitangents and the congruence of inflectional tangents. We give new proofs for the bidegrees of the secant, bitangent and inflectional congruences, using geometric techniques such as duality, polar loci and projections. We also study the singularities of these congruences.