The implicit equation of a multigraded hypersurface

Abstract

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