The implicit equation of a multigraded hypersurface

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 Ree...

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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:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00218693_v348_n1_p381_Botbol http://hdl.handle.net/20.500.12110/paper_00218693_v348_n1_p381_Botbol
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_00218693_v348_n1_p381_Botbol
record_format	dspace
spelling	paper:paper_00218693_v348_n1_p381_Botbol2023-06-08T14:42:26Z The implicit equation of a multigraded hypersurface Approximation complex Castelnuovo-Mumford regularity Elimination theory Graded algebra Graded ring Hypersurfaces Implicit equation Implicitization Koszul complex Multigraded algebra Multigraded ring Resultant Toric variety 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. 2011 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00218693_v348_n1_p381_Botbol http://hdl.handle.net/20.500.12110/paper_00218693_v348_n1_p381_Botbol
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (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 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.
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	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00218693_v348_n1_p381_Botbol http://hdl.handle.net/20.500.12110/paper_00218693_v348_n1_p381_Botbol
_version_	1768545403445182464

The implicit equation of a multigraded hypersurface

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