The implicit equation of a multigraded hypersurface

• Autores: Botbol, N.
• Año de publicación: 2011
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: J. Algebra 2011;348(1):381-401
• Materia: Approximation complex; Castelnuovo-Mumford regularity; Elimination theory; Graded algebra; Graded ring; Hypersurfaces; Implicit equation; Implicitization; Koszul complex; Multigraded algebra; Multigraded ring; Resultant; Toric variety
• Institución: Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales

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.