A focus on focal surfaces

TL;DR

This paper systematically studies the focal surfaces of line congruences in projective space, revealing their invariants and uncovering unexpected components in specific congruences, with conjectures about their uniqueness.

Contribution

It provides a comprehensive analysis of focal surfaces using differential and intersection theory, identifying new properties and components in special congruences.

Findings

Focal surfaces' invariants are explicitly computed.

Uncovering unexpected components in congruences of chords, bitangents, and flexes.

Conjecture on the uniqueness of these components in such congruences.

Abstract

We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal surface (degree, class, class of its hyperplane section, sectional genus and degrees of the nodal and cuspidal curve). We study in particular the congruences of chords to a smooth curve and the congruences of bitangents or flexes to a smooth surface. We find that they possess unexpected components in their focal surface, and conjecture that they are the only ones with this property.