On the monodromy of the inflection points of plane curves

TL;DR

This paper determines the monodromy group of inflection points of plane curves, showing it is the symmetric group for degree d≥4 and characterizing it explicitly for degree 3, revealing deep geometric symmetry properties.

Contribution

It establishes the monodromy group as the symmetric group for degree d≥4 and describes its structure for degree 3, advancing understanding of inflection point symmetries.

Findings

Monodromy group is the symmetric group for d≥4.

For d=3, the monodromy group is the projective symmetry group of the Fermat curve.

Provides explicit descriptions of monodromy groups for plane curves.

Abstract

We prove that the monodromy group of the inflection points of plane curves of degree $d$ is the symmetric group $\mathbb S_{3d(d-2)}$ for $d\geq 4$ and in the case $d=3$ this group is the group of the projective transformations of $\mathbb P^2$ leaving invariant the nine inflection points of the Fermat curve of degree three.