The implicit equation of a multigraded hypersurface

In this article we analyze the implicitization problem of the image of a rational map $\varphi :X-->P^n$, with $T$ a toric variety of dimension $n-1$ defined by its Cox ring $R$. Let $I:=(f_0,\mathrm{.}\mathrm{.}\mathrm{.},f_n)$ be $n+1$ homogeneous elements of $R$. We blow-up the base locus of $\varphi$, $V(I)$, and we approximate the Rees algebra $Ree{s}_R(I)$ of this blow-up by the symmetric algebra $Sy{m}_R(I)$. We provide under suitable assumptions, resolutions $\mathbb{Z}\mathrm{.}$ for $Sy{m}_R(I)$ graded by the torus-invariant divisor group of $X$, $Cl(X)$, such that the determinant of a graded strand, $\det ((\mathbb{Z}\mathrm{.}{)}_{\mu })$, gives a multiple of the implicit equation, for suitable $\mu \in Cl(X)$. Indeed, we compute a region in $Cl(X)$ which depends on the regularity of $Sy{m}_R(I)$ where to choose $\mu$. We also give a geometrical interpretation of the possible other factors appearing in $\det ((\mathbb{Z}\mathrm{.}{)}_{\mu })$. A very detailed description is given when $X$ is a multiprojective space.