On the variety of the inflection points of plane cubic curves

TL;DR

This paper studies the geometric properties of the inflection points of plane cubic curves, including their monodromy, normalization, and irregularity, revealing new insights into their algebraic and topological structure.

Contribution

It provides a detailed analysis of the variety of inflection points, including monodromy groups, normalizations, and irregularity, for general and singular cubic curves.

Findings

Description of local monodromy groups near singular cubics

Normalizations of surfaces of inflection points in linear systems

Proof of vanishing irregularity of the associated smooth manifold

Abstract

In the paper, we investigate properties of the nine-dimensional variety of the inflection points of the plane cubic curves. The description of local monodromy groups of the set of inflection points near singular cubic curves is given. Also, it is given a detailed description of the normalizations of the surfaces of the inflection points of plane cubic curves belonging to general two-dimensional linear systems of cubic curves, The vanishing of the irregularity a smooth manifold birationally isomorphic to the variety of the inflection points of the plane cubic curves is proved.