The implicit equation of a multigraded hypersurface
In this article we analyze the implicitization problem of the image of a rational map φ : X 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 Re...
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2011 |
| País: | Argentina |
| Recursos: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/15925 |
| Acesso em linha: | http://hdl.handle.net/11336/15925 |
| Access Level: | acceso abierto |
| Palavra-chave: | Implicitization Multigraded Algebra Representation Matrices Approximation Complex Castelnuovo–Mumford Regularity https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Resumo: | In this article we analyze the implicitization problem of the image of a rational map φ : X 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. |
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