On the equations of the moving curve ideal of a rational algebraic plane curve

TL;DR

This paper investigates the algebraic structure of the moving curve ideal of a rational plane algebraic curve, providing explicit descriptions of adjoint pencils and generators of the Rees algebra through elimination theory.

Contribution

It offers new explicit formulas for adjoint pencils and generators of the Rees algebra for rational plane curves, advancing understanding of their algebraic equations.

Findings

Explicit adjoint pencils are described via determinants.

Generators of the Rees algebra are explicitly constructed.

Detailed analysis of the elimination ideal of regular sequences.

Abstract

Given a parametrization of a rational plane algebraic curve C, some explicit adjoint pencils on C are described in terms of determinants. Moreover, some generators of the Rees algebra associated to this parametrization are presented. The main ingredient developed in this paper is a detailed study of the elimination ideal of two homogeneous polynomials in two homogeneous variables that form a regular sequence.