Freeness and invariants of rational plane curves

TL;DR

This paper investigates invariants of rational plane curves through their parameterizations, establishing new characterizations of cuspidal curves, linking freeness to Hilbert functions, and providing methods to compute Tjurina numbers directly from parameterizations.

Contribution

It introduces novel methods to analyze rational plane curves' invariants using parameterizations, including characterizations of cuspidal curves and criteria for freeness based on Hilbert functions.

Findings

Characterization of rational cuspidal curves via discriminant and dual curve relations.

Freeness of rational curves can be tested through Hilbert functions of kernel of a canonical map.

Tjurina number can be computed directly from a parameterization without explicit equations.

Abstract

Given a parameterization $\phi$ of a rational plane curve C, we study some invariants of C via $\phi$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $\phi$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $\phi$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational curve can be computed directly from one of its parameterization, without relying on the computation of an equation of C.