On the base point locus of surface parametrizations: formulas and consequences

TL;DR

This paper investigates the base point locus of surface parametrizations, providing formulas to compute multiplicities and degrees, and explores how reparametrizations influence the surface degree and base point structure.

Contribution

It introduces a new formula linking base point multiplicity to the content of a univariate resultant and extends degree formulas to rational maps of the projective plane.

Findings

Base point multiplicity expressed via univariate resultants

Degree formula relating surface degree, parametrization degree, and base points

Reparametrization degree affected by base points presence

Abstract

This paper shows that the multiplicity of the base points locus of a projective rational surface parametrization can be expressed as the degree of the content of a univariate resultant. As a consequence, we get a new proof of the degree formula relating the degree of the surface, the degree of the parametrization, the base points multiplicity, and the degree of the rational map induced by the parametrization. In addition, we extend both formulas to the case of dominant rational maps of the projective plane and describe how the base point loci of a parametrization and its reparametrizations are related. As an application of these results, we explore how the degree of a surface reparametrization is affected by the presence of base points.