Projective classification of jets of surfaces in 3-space

TL;DR

This paper provides a local classification of smooth projective surfaces in 3-space based on projective transformations and singularity types of central projections, linking Monge forms with bifurcations of key geometric curves.

Contribution

It introduces a classification scheme for surfaces in 3-space considering singularity types up to codimension 4 and relates it to bifurcations of important geometric features.

Findings

Classification of surfaces based on singularity types

Relation between Monge forms and bifurcations of parabolic and flecnodal curves

Framework for analyzing surface projections in projective geometry

Abstract

We present a local classification of smooth projective surfaces in 3-space via projective transformations in accordance with singularity types of central projections up to codimension 4. We also discuss relations between our classification of Monge forms and bifurcations of parabolic curves and flecnodal curves.