Singularities of plane rational curves via projections

TL;DR

This paper develops a geometric framework using projections from rational normal curves to analyze and classify singularities of plane rational curves, providing algorithms to extract singularity information from associated schemes.

Contribution

It introduces a novel approach linking projections and secant varieties to encode singularities via schemes, with algorithms for their analysis.

Findings

Schemes $X_k$ encode singularities of multiplicity ≥ k.

Criterion to distinguish cuspidal curves from others.

Algorithms to determine singularity types from schemes.

Abstract

We consider the parameterization ${\mathbf{f}}=(f_0,f_1,f_2)$ of a plane rational curve $C$ of degree $n$, and we want to study the singularities of $C$ via such parameterization. We do this by using the projection from the rational normal curve $C_n\subset \mathbb{P}^n$ to $C$ and its interplay with the secant varieties to $C_n$. In particular, we define via ${\mathbf{f}}$ certain 0-dimensional schemes $X_k\subset \mathbb{P}^k$, $2\leq k\leq (n-1)$, which encode all information on the singularities of multiplicity $\geq k$ of $C$ (e.g. using $X_2$ we can give a criterion to determine whether $C$ is a cuspidal curve or has only ordinary singularities). We give a series of algorithms which allow to get info about the singularities from such schemes.