The implicit equation of a multigraded hypersurface

Abstract

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.