The implicit equation of a multigraded hypersurface - Botbol, N. (2011)
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.