Binary differential equations at parabolic and umbilical points for $2$-parameter families of surfaces

TL;DR

This paper classifies the local topological types of binary differential equations related to asymptotic curves at special points on surfaces in projective space, providing insights into bifurcations and the behavior of the flecnodal curve.

Contribution

It offers a projective classification of Monge forms and compares it with existing BDE classifications, identifying new bifurcation diagrams for generic surface families.

Findings

Classification of bifurcations of the parabolic curve

New bifurcation diagrams for typical examples

Analysis of the flecnodal curve behavior

Abstract

We determine local topological types of binary differential equations of asymptotic curves at parabolic and flat umbilical points for generic $2$-parameter families of surfaces in $\mathbb P^3$ by comparing our projective classification of Monge forms and classification of general BDE obtained by Tari and Oliver. In particular, generic bifurcations of the parabolic curve are classified. The flecnodal curve is also examined by direct computations, and we present new bifurcation diagrams in typical examples.