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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Published in | Communications in mathematics and statistics Vol. 10; no. 4; pp. 757 - 783 |
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Main Authors | , , |
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01.12.2022
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Abstract | 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. |
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AbstractList | Abstract
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. 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. |
Author | Cox, David A. Pérez-Díaz, Sonia Sendra, J. Rafael |
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Cites_doi | 10.1016/j.jpaa.2004.02.011 10.1016/j.jsc.2004.09.007 10.1017/CBO9780511608681 10.1142/S0219498803000489 10.1007/s11786-013-0139-8 10.1007/978-3-7091-6571-3 10.1016/j.jsc.2004.02.004 10.1016/j.cam.2017.07.023 10.1007/s00209-006-0941-y 10.1016/j.jalgebra.2017.12.028 10.1016/S0022-4049(98)00078-4 10.1016/j.jsc.2007.10.001 10.1090/mcom/3193 10.1016/j.jsc.2005.01.003 10.1145/780506.780536 10.1109/38.56295 10.1090/conm/334/05980 10.1145/2608628.2608635 10.1007/978-3-540-73725-4 10.1090/conm/286/04751 10.1090/conm/334/05979 |
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Keywords | Base point 14J29 14Q10 Hilbert–Samuel multiplicity Surface parametrization Reparametrization 14J70 Surface degree Parametrization degree |
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References | Pérez-Díaz, Sendra (CR15) 2008; 43 Walker (CR24) 1950 Bruns, Herzog (CR3) 1998 CR18 Schicho (CR19) 2006; 254 Chen, Cox, Liu (CR5) 2005; 39 CR11 CR10 Pérez-Díaz, Sendra (CR14) 2004; 193 Busé, Cox, D’Andrea (CR2) 2003; 2 Jia, Shi, Chen (CR13) 2018; 329 Wang (CR25) 2004; 38 Pérez-Díaz, Sendra (CR16) 2013; 7 CR6 CR8 CR7 CR9 CR27 Winkler (CR26) 1996 Harris (CR12) 1995 CR23 CR21 CR20 Schicho (CR17) 1999; 145 Sendra, Sevilla, Villarino (CR22) 2017; 86 Adkins, Hoffman, Wang (CR1) 2005; 39 Caravantes, Sendra, Sevilla, Villarino (CR4) 2018; 501 257_CR10 F Chen (257_CR5) 2005; 39 257_CR11 D Wang (257_CR25) 2004; 38 J Harris (257_CR12) 1995 W Bruns (257_CR3) 1998 WA Adkins (257_CR1) 2005; 39 S Pérez-Díaz (257_CR14) 2004; 193 J Caravantes (257_CR4) 2018; 501 J Schicho (257_CR19) 2006; 254 F Winkler (257_CR26) 1996 257_CR21 257_CR20 257_CR23 257_CR9 257_CR8 257_CR7 257_CR27 257_CR6 X Jia (257_CR13) 2018; 329 JR Sendra (257_CR22) 2017; 86 RJ Walker (257_CR24) 1950 S Pérez-Díaz (257_CR15) 2008; 43 J Schicho (257_CR17) 1999; 145 257_CR18 L Busé (257_CR2) 2003; 2 S Pérez-Díaz (257_CR16) 2013; 7 |
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Title | On the Base Point Locus of Surface Parametrizations: Formulas and Consequences |
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