The implicit equation of a multigraded hypersurface

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In this article we analyze the implicitization problem of the image of a rational map φ:X[U+21E2]Pn, with X a toric variety of dimension n-1 defined by its Cox ring R. Let I:=(f0, ..., fn) be n+1 homogeneous elements of R. We blow-up the base locus of φ, V(I), and we approximate the...

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Autor principal:	Botbol, N.
Formato:	Artículo publishedVersion
Publicado:	2011
Materias:	Approximation complex Castelnuovo-Mumford regularity Elimination theory Graded algebra Graded ring Hypersurfaces Implicit equation Implicitization Koszul complex Multigraded algebra Multigraded ring Resultant Toric variety
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_00218693_v348_n1_p381_Botbol https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00218693_v348_n1_p381_Botbol_oai
Aporte de:	Repositorio Digital de la Universidad de Buenos Aires (UBA) de Universidad de Buenos Aires

id	I28-R145-paper_00218693_v348_n1_p381_Botbol_oai
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spelling	I28-R145-paper_00218693_v348_n1_p381_Botbol_oai2024-08-16 Botbol, N. 2011 In this article we analyze the implicitization problem of the image of a rational map φ:X[U+21E2]Pn, with X a toric variety of dimension n-1 defined by its Cox ring R. Let I:=(f0, ..., fn) be n+1 homogeneous elements of R. We blow-up the base locus of φ, V(I), and we approximate the Rees algebra ReesR(I) of this blow-up by the symmetric algebra SymR(I). We provide under suitable assumptions, resolutions Z for SymR(I) graded by the divisor group of X, Cl(X), such that the determinant of a graded strand, det((Z)μ), gives a multiple of the implicit equation, for suitable μ∈Cl(X). Indeed, we compute a region in Cl(X) which depends on the regularity of SymR(I) where to choose μ. We also give a geometrical interpretation of the possible other factors appearing in det((Z)μ). A very detailed description is given when X is a multiprojective space. © 2011 Elsevier Inc. application/pdf http://hdl.handle.net/20.500.12110/paper_00218693_v348_n1_p381_Botbol info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar J. Algebra 2011;348(1):381-401 Approximation complex Castelnuovo-Mumford regularity Elimination theory Graded algebra Graded ring Hypersurfaces Implicit equation Implicitization Koszul complex Multigraded algebra Multigraded ring Resultant Toric variety The implicit equation of a multigraded hypersurface info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00218693_v348_n1_p381_Botbol_oai
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-145
collection	Repositorio Digital de la Universidad de Buenos Aires (UBA)
topic	Approximation complex Castelnuovo-Mumford regularity Elimination theory Graded algebra Graded ring Hypersurfaces Implicit equation Implicitization Koszul complex Multigraded algebra Multigraded ring Resultant Toric variety
spellingShingle	Approximation complex Castelnuovo-Mumford regularity Elimination theory Graded algebra Graded ring Hypersurfaces Implicit equation Implicitization Koszul complex Multigraded algebra Multigraded ring Resultant Toric variety Botbol, N. The implicit equation of a multigraded hypersurface
topic_facet	Approximation complex Castelnuovo-Mumford regularity Elimination theory Graded algebra Graded ring Hypersurfaces Implicit equation Implicitization Koszul complex Multigraded algebra Multigraded ring Resultant Toric variety
description	In this article we analyze the implicitization problem of the image of a rational map φ:X[U+21E2]Pn, with X a toric variety of dimension n-1 defined by its Cox ring R. Let I:=(f0, ..., fn) be n+1 homogeneous elements of R. We blow-up the base locus of φ, V(I), and we approximate the Rees algebra ReesR(I) of this blow-up by the symmetric algebra SymR(I). We provide under suitable assumptions, resolutions Z for SymR(I) graded by the divisor group of X, Cl(X), such that the determinant of a graded strand, det((Z)μ), gives a multiple of the implicit equation, for suitable μ∈Cl(X). Indeed, we compute a region in Cl(X) which depends on the regularity of SymR(I) where to choose μ. We also give a geometrical interpretation of the possible other factors appearing in det((Z)μ). A very detailed description is given when X is a multiprojective space. © 2011 Elsevier Inc.
format	Artículo Artículo publishedVersion
author	Botbol, N.
author_facet	Botbol, N.
author_sort	Botbol, N.
title	The implicit equation of a multigraded hypersurface
title_short	The implicit equation of a multigraded hypersurface
title_full	The implicit equation of a multigraded hypersurface
title_fullStr	The implicit equation of a multigraded hypersurface
title_full_unstemmed	The implicit equation of a multigraded hypersurface
title_sort	implicit equation of a multigraded hypersurface
publishDate	2011
url	http://hdl.handle.net/20.500.12110/paper_00218693_v348_n1_p381_Botbol https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00218693_v348_n1_p381_Botbol_oai
work_keys_str_mv	AT botboln theimplicitequationofamultigradedhypersurface AT botboln implicitequationofamultigradedhypersurface
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The implicit equation of a multigraded hypersurface

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