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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| Formato: | Artículo publishedVersion |
| Lenguaje: | Inglés |
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2011
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00218693_v348_n1_p381_Botbol |
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paperaa:paper_00218693_v348_n1_p381_Botbol2023-06-12T16:42:28Z The implicit equation of a multigraded hypersurface J. Algebra 2011;348(1):381-401 Botbol, N. 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 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion application/pdf eng info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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) |
| language |
Inglés |
| orig_language_str_mv |
eng |
| 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 |
| work_keys_str_mv |
AT botboln theimplicitequationofamultigradedhypersurface AT botboln implicitequationofamultigradedhypersurface |
| _version_ |
1769810079037521920 |