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 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.