Deformations of plane curves and Jacobian syzygies

TL;DR

This paper explores the relationship between deformations of complex plane curves and Jacobian syzygies, providing examples and conjectures, especially focusing on rational cuspidal curves.

Contribution

It establishes a connection between curve deformations and Jacobian syzygies, offering new insights and conjectures in the study of rational cuspidal curves.

Findings

Relation between equianalytic and equisingular deformations and Jacobian syzygies

Examples illustrating the theory with rational cuspidal curves

Conjectures on the behavior of these curves

Abstract

We relate the equianalytic and the equisingular deformations of a reduced complex plane curve to the Jacobian syzygies of its defining equation. Several examples and conjectures involving rational cuspidal curves are discussed.