Determining plane curve singularities from its polars

TL;DR

This paper presents a method to determine the singularity type of a plane curve by analyzing its polar curves and their base points, providing explicit constructions and classifications.

Contribution

It introduces a direct construction of singular points from polar base points and determines the equisingularity class from polar data and shared non-singular points.

Findings

Explicit construction of singular points from polar base points

Determination of equisingularity class from polar curves

Connection between polar genericity and singularity classification

Abstract

This paper addresses a very classical topic that goes back at least to Pl\"ucker: how to understand a plane curve singularity using its polar curves. Here, we explicitly construct the singular points of a plane curve singularity directly from the weighted cluster of base points of its polars. In particular, we determine the equisingularity class (or topological equivalence class) of a germ of plane curve from the equisingularity class of generic polars and combinatorial data about the non-singular points shared by them.