# Secants, bitangents, and their congruences

## TL;DR

This paper investigates the geometric properties and singularities of congruences of lines related to space curves and surfaces in projective 3-space, providing new proofs for their bidegrees and analyzing their singular loci.

## Contribution

It introduces new geometric proofs for the bidegrees of secant, bitangent, and inflectional congruences, and characterizes their singular loci using advanced algebraic geometry techniques.

## Key findings

- Singular locus of the Chow hypersurface contains the secant congruence.
- Singular locus of the Hurwitz hypersurface includes bitangent and inflectional tangent congruences.
- New geometric proofs for bidegrees of these congruences.

## Abstract

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.