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Abstract:

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.

Registro:

Documento: Artículo
Título:The implicit equation of a multigraded hypersurface
Autor:Botbol, N.
Filiación:Departamento de Matemática, FCEN, Universidad de Buenos Aires, Argentina
Institut de Mathématiques de Jussieu, Université de P. et M. Curie, Paris VI, France
Palabras clave:Approximation complex; Castelnuovo-Mumford regularity; Elimination theory; Graded algebra; Graded ring; Hypersurfaces; Implicit equation; Implicitization; Koszul complex; Multigraded algebra; Multigraded ring; Resultant; Toric variety
Año:2011
Volumen:348
Número:1
Página de inicio:381
Página de fin:401
DOI: http://dx.doi.org/10.1016/j.jalgebra.2011.09.019
Título revista:Journal of Algebra
Título revista abreviado:J. Algebra
ISSN:00218693
CODEN:JALGA
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_00218693_v348_n1_p381_Botbol.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00218693_v348_n1_p381_Botbol

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Citas:

---------- APA ----------
(2011) . The implicit equation of a multigraded hypersurface. Journal of Algebra, 348(1), 381-401.
http://dx.doi.org/10.1016/j.jalgebra.2011.09.019
---------- CHICAGO ----------
Botbol, N. "The implicit equation of a multigraded hypersurface" . Journal of Algebra 348, no. 1 (2011) : 381-401.
http://dx.doi.org/10.1016/j.jalgebra.2011.09.019
---------- MLA ----------
Botbol, N. "The implicit equation of a multigraded hypersurface" . Journal of Algebra, vol. 348, no. 1, 2011, pp. 381-401.
http://dx.doi.org/10.1016/j.jalgebra.2011.09.019
---------- VANCOUVER ----------
Botbol, N. The implicit equation of a multigraded hypersurface. J. Algebra. 2011;348(1):381-401.
http://dx.doi.org/10.1016/j.jalgebra.2011.09.019