The topology of the generic polar curve and the Zariski invariant for branches of genus one

TL;DR

This paper investigates the topological structure of generic polar curves associated with genus one plane complex branches, linking it to their semigroup of values and Zariski invariant, and refines previous results by Casas-Alvero.

Contribution

It provides a detailed analysis of how the topological type of the polar curve depends on algebraic invariants, improving the understanding of their classification.

Findings

Filtered the topological types of polar curves using Zariski invariants.

Established a relationship between semigroup values and polar curve topology.

Refined Casas-Alvero's results on genus one branches.

Abstract

We study, for plane complex branches of genus one, the topological type of its generic polar curve, as a function of the semigroup of values and the Zariski invariant of the branch. We improve some results given by Casas-Alvero in 2023, since we filter the topological type fixed for the branch by the possible values of Zariski invariants.