Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field

Autores
Cafure, Antonio Artemio; Matera, Guillermo; Privitelli, Melina Lorena
Año de publicación
2015
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Let V ⊂ double-struck Pn(double-struck F¯q) be a complete intersection defined over a finite field double-struck Fq of dimension r and singular locus of dimension at most s, and let π :V → double-struck Ps+1 (double-struck F¯q) be a generic linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning π, namely an explicit upper bound of the degree of a proper Zariski closed subset of double-struck Ps+1(double-struck F¯q) which contains all the points defining singular fibers of π. For this purpose we make use of the concept of polar variety associated with the set of exceptional points of π. As a consequence, we obtain results of existence of smooth rational points of V, that is, conditions on q which imply that V has a smooth double-struck Fq-rational point. Finally, for s = r - 2 and s = r - 3 we estimate the number of double-struck Fq-rational points and smooth double-struck Fq-rational points of V.
Fil: Cafure, Antonio Artemio. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Ciclo Básico Común; Argentina
Fil: Matera, Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Privitelli, Melina Lorena. Universidad de Buenos Aires. Ciclo Básico Común; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina
Materia
Bertini Smoothness Theorem
Deligne Estimate
Hooley-Katz Estimate
Multihomogeneous BÉZout Theorem
Polar Varieties
Rational Points
Singular Locus
Varieties Over Finite Fields
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/37729
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/37729
network_acronym_str CONICETDig
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network_name_str CONICET Digital (CONICET)
spelling Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite fieldCafure, Antonio ArtemioMatera, GuillermoPrivitelli, Melina LorenaBertini Smoothness TheoremDeligne EstimateHooley-Katz EstimateMultihomogeneous BÉZout TheoremPolar VarietiesRational PointsSingular LocusVarieties Over Finite Fieldshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Let V ⊂ double-struck Pn(double-struck F¯q) be a complete intersection defined over a finite field double-struck Fq of dimension r and singular locus of dimension at most s, and let π :V → double-struck Ps+1 (double-struck F¯q) be a generic linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning π, namely an explicit upper bound of the degree of a proper Zariski closed subset of double-struck Ps+1(double-struck F¯q) which contains all the points defining singular fibers of π. For this purpose we make use of the concept of polar variety associated with the set of exceptional points of π. As a consequence, we obtain results of existence of smooth rational points of V, that is, conditions on q which imply that V has a smooth double-struck Fq-rational point. Finally, for s = r - 2 and s = r - 3 we estimate the number of double-struck Fq-rational points and smooth double-struck Fq-rational points of V.Fil: Cafure, Antonio Artemio. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Ciclo Básico Común; ArgentinaFil: Matera, Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Privitelli, Melina Lorena. Universidad de Buenos Aires. Ciclo Básico Común; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaElsevier2015-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/37729Cafure, Antonio Artemio; Matera, Guillermo; Privitelli, Melina Lorena; Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field; Elsevier; Finite Fields and Their Applications; 31; 1-2015; 42-831071-5797CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S1071579714001051info:eu-repo/semantics/altIdentifier/doi/10.1016/j.ffa.2014.09.002info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:15:09Zoai:ri.conicet.gov.ar:11336/37729instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:15:09.995CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field
title Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field
spellingShingle Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field
Cafure, Antonio Artemio
Bertini Smoothness Theorem
Deligne Estimate
Hooley-Katz Estimate
Multihomogeneous BÉZout Theorem
Polar Varieties
Rational Points
Singular Locus
Varieties Over Finite Fields
title_short Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field
title_full Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field
title_fullStr Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field
title_full_unstemmed Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field
title_sort Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field
dc.creator.none.fl_str_mv Cafure, Antonio Artemio
Matera, Guillermo
Privitelli, Melina Lorena
author Cafure, Antonio Artemio
author_facet Cafure, Antonio Artemio
Matera, Guillermo
Privitelli, Melina Lorena
author_role author
author2 Matera, Guillermo
Privitelli, Melina Lorena
author2_role author
author
dc.subject.none.fl_str_mv Bertini Smoothness Theorem
Deligne Estimate
Hooley-Katz Estimate
Multihomogeneous BÉZout Theorem
Polar Varieties
Rational Points
Singular Locus
Varieties Over Finite Fields
topic Bertini Smoothness Theorem
Deligne Estimate
Hooley-Katz Estimate
Multihomogeneous BÉZout Theorem
Polar Varieties
Rational Points
Singular Locus
Varieties Over Finite Fields
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Let V ⊂ double-struck Pn(double-struck F¯q) be a complete intersection defined over a finite field double-struck Fq of dimension r and singular locus of dimension at most s, and let π :V → double-struck Ps+1 (double-struck F¯q) be a generic linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning π, namely an explicit upper bound of the degree of a proper Zariski closed subset of double-struck Ps+1(double-struck F¯q) which contains all the points defining singular fibers of π. For this purpose we make use of the concept of polar variety associated with the set of exceptional points of π. As a consequence, we obtain results of existence of smooth rational points of V, that is, conditions on q which imply that V has a smooth double-struck Fq-rational point. Finally, for s = r - 2 and s = r - 3 we estimate the number of double-struck Fq-rational points and smooth double-struck Fq-rational points of V.
Fil: Cafure, Antonio Artemio. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Ciclo Básico Común; Argentina
Fil: Matera, Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Privitelli, Melina Lorena. Universidad de Buenos Aires. Ciclo Básico Común; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina
description Let V ⊂ double-struck Pn(double-struck F¯q) be a complete intersection defined over a finite field double-struck Fq of dimension r and singular locus of dimension at most s, and let π :V → double-struck Ps+1 (double-struck F¯q) be a generic linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning π, namely an explicit upper bound of the degree of a proper Zariski closed subset of double-struck Ps+1(double-struck F¯q) which contains all the points defining singular fibers of π. For this purpose we make use of the concept of polar variety associated with the set of exceptional points of π. As a consequence, we obtain results of existence of smooth rational points of V, that is, conditions on q which imply that V has a smooth double-struck Fq-rational point. Finally, for s = r - 2 and s = r - 3 we estimate the number of double-struck Fq-rational points and smooth double-struck Fq-rational points of V.
publishDate 2015
dc.date.none.fl_str_mv 2015-01
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/37729
Cafure, Antonio Artemio; Matera, Guillermo; Privitelli, Melina Lorena; Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field; Elsevier; Finite Fields and Their Applications; 31; 1-2015; 42-83
1071-5797
CONICET Digital
CONICET
url http://hdl.handle.net/11336/37729
identifier_str_mv Cafure, Antonio Artemio; Matera, Guillermo; Privitelli, Melina Lorena; Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field; Elsevier; Finite Fields and Their Applications; 31; 1-2015; 42-83
1071-5797
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S1071579714001051
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.ffa.2014.09.002
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.070432

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