Degree and birationality of multi-graded rational maps

We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is...

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Authors: Busé, Laurent, Cid Ruiz, Yairon, D'Andrea, Carlos, 1973-
Format: article
Status:Versión aceptada para publicación
Publication Date:2020
Country:España
Institution:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repository:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2445/196340
Online Access:https://hdl.handle.net/2445/196340
Access Level:Open access
Keyword:Anells commutatius
Geometria biracional
Homologia
Commutative rings
Birational geometry
Homology
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spelling Degree and birationality of multi-graded rational mapsBusé, LaurentCid Ruiz, YaironD'Andrea, Carlos, 1973-Anells commutatiusGeometria biracionalHomologiaCommutative ringsBirational geometryHomologyWe give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the saturated special fiber ring, which turns out to be a fundamental tool to analyze the degree of a rational map. We also provide a very effective birationality criterion and a complete description of the equations of the associated Rees algebra of a particular class of plane rational maps.Oxford University Press2023202320202023info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersion45 p.application/pdfhttps://hdl.handle.net/2445/196340Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésVersió postprint del document publicat a: https://doi.org/10.1112/plms.12336Proceedings of the London Mathematical Society, 2020, vol. 121, num. 4, p. 743-787https://doi.org/10.1112/plms.12336(c) London Mathematical Society, 2020info:eu-repo/semantics/openAccessoai:recercat.cat:2445/1963402026-05-29T05:05:01Z
dc.title.none.fl_str_mv Degree and birationality of multi-graded rational maps
title Degree and birationality of multi-graded rational maps
spellingShingle Degree and birationality of multi-graded rational maps
Busé, Laurent
Anells commutatius
Geometria biracional
Homologia
Commutative rings
Birational geometry
Homology
title_short Degree and birationality of multi-graded rational maps
title_full Degree and birationality of multi-graded rational maps
title_fullStr Degree and birationality of multi-graded rational maps
title_full_unstemmed Degree and birationality of multi-graded rational maps
title_sort Degree and birationality of multi-graded rational maps
dc.creator.none.fl_str_mv Busé, Laurent
Cid Ruiz, Yairon
D'Andrea, Carlos, 1973-
author Busé, Laurent
author_facet Busé, Laurent
Cid Ruiz, Yairon
D'Andrea, Carlos, 1973-
author_role author
author2 Cid Ruiz, Yairon
D'Andrea, Carlos, 1973-
author2_role author
author
dc.subject.none.fl_str_mv Anells commutatius
Geometria biracional
Homologia
Commutative rings
Birational geometry
Homology
topic Anells commutatius
Geometria biracional
Homologia
Commutative rings
Birational geometry
Homology
description We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the saturated special fiber ring, which turns out to be a fundamental tool to analyze the degree of a rational map. We also provide a very effective birationality criterion and a complete description of the equations of the associated Rees algebra of a particular class of plane rational maps.
publishDate 2020
dc.date.none.fl_str_mv 2020
2023
2023
2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/196340
url https://hdl.handle.net/2445/196340
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Versió postprint del document publicat a: https://doi.org/10.1112/plms.12336
Proceedings of the London Mathematical Society, 2020, vol. 121, num. 4, p. 743-787
https://doi.org/10.1112/plms.12336
dc.rights.none.fl_str_mv (c) London Mathematical Society, 2020
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) London Mathematical Society, 2020
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 45 p.
application/pdf
dc.publisher.none.fl_str_mv Oxford University Press
publisher.none.fl_str_mv Oxford University Press
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,228081

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