Remarks on double points of plane curves

TL;DR

This paper investigates the relationship between double points on plane curves and associated curvilinear subschemes, introduces an algorithm for analyzing these singularities, and explores constructing rational curves with specific singularities.

Contribution

It presents a new algorithm for studying double points on plane curves and offers methods to construct rational curves with prescribed singularities.

Findings

Established a link between double points and curvilinear subschemes

Developed an algorithm for analyzing double points via osculating curves

Proposed a method to create rational curves with specific singularities

Abstract

We study the relation between the type of a double point of a plane curve and the curvilinear 0-dimensional subschemes of the curve at the point. An Algorithm related to a classical procedure for the study of double points via osculating curves is described and proved. Eventually we look for a way to create examples of rational plane curves with given singularities $A_s$.