On the structure of mu-classes

TL;DR

This paper proves a conjecture about the limits of rational plane curves of a given degree and class, enabling explicit descriptions of their parametrization varieties.

Contribution

It establishes that rational plane curves of degree n and class re limits of those with class nd , confirming a conjecture and providing a new description of parametrization varieties.

Findings

Proves the conjecture for  nd .

Describes the variety of parametrizations explicitly.

Enables understanding of limits of rational plane curves.

Abstract

We prove that, if \mu<\lfloor n/2\rfloor, then every rational plane curve of degree n and class \mu is a limit of parametrizations of the same degree and class \mu+1. This property was conjectured in D.Cox, T.Sederberg,F.Chen's paper: "The moving line ideal basis of planar rational curves" (Comput. Aided Geom. Des. 15 (1998), 803-827), and its validity allows an explicit description of the variety of parametrizations of degree n and class \mu, for all (n,\mu).