Normalization of congruence of bitangents to a hypersurface in $\mathbb P^3$

TL;DR

This paper studies the normalization of the surface of bitangent lines to hypersurfaces in projective 3-space, providing criteria for smoothness and describing degenerations within Lefschetz pencils.

Contribution

It introduces the concept of Fano congruence of bitangents and offers explicit criteria and descriptions for their smoothness and degenerations.

Findings

Criteria for smoothness of the Fano congruence of bitangents.

Explicit descriptions of degenerations in Lefschetz pencils.

Analysis of the normalization process for congruences in Grassmannians.

Abstract

A congruence is a surface in the Grassmannian ${\rm Gr}(2, 4)$. In this paper, we consider the normalization of congruence of bitangents to a hypersurface in $\mathbb P^3$. We call it the Fano congruence of bitangents. We give a criterion for smoothness of the Fano congruence of bitangents and describe explicitly their degenerations in a general Lefschetz pencil in the space of hypersurfaces in $\mathbb P^3$.