On the Base Point Locus of Surface Parametrizations: Formulas and Consequences

This paper shows that the multiplicity of the base point 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 pa...

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Bibliographic Details
Published inCommunications in mathematics and statistics Vol. 10; no. 4; pp. 757 - 783
Main Authors Cox, David A., Pérez-Díaz, Sonia, Sendra, J. Rafael
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2022
Springer Nature B.V
Subjects
Loci
Mathematics
Mathematics and Statistics
Parameterization
Statistics
Base point
14J29
14Q10
Hilbert–Samuel multiplicity
Surface parametrization
Reparametrization
14J70
Surface degree
Parametrization degree
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Summary:This paper shows that the multiplicity of the base point 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 point 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.
ISSN:2194-6701
2194-671X
DOI:10.1007/s40304-021-00257-4