On the Jacobian Scheme of a plane curve

TL;DR

This paper investigates the Jacobian scheme of plane algebraic curves at ordinary singularities, providing geometric characterizations, computing Tjurina numbers, and offering an algorithm for classifying double points using algebraic methods.

Contribution

It introduces a geometric characterization of the Jacobian scheme at ordinary singularities, computes minimal Tjurina numbers for certain curves, and presents an algorithm for classifying double points.

Findings

Characterization of the Jacobian scheme at ordinary singularities

Tjurina number reaches minimum at certain curves

Algorithm for analytic type classification of double points

Abstract

We study the Jacobian scheme of a plane algebraic curve at an ordinary singularity, characterizing it through a geometric property. We compute the Tjurina number for a family of curves at an ordinary singularity showing that it reaches the minimum possible value, using very elementary methods, essentially Gr\"obner basis. We give an algorithm that gives the analytic type of a double point using the algebraic version of the Mather-Yau Theorem.