The implicit equation of a multigraded hypersurface

TL;DR

This paper studies the implicitization of rational maps from toric varieties to projective space, providing algebraic resolutions and geometric interpretations to compute implicit equations, especially for multiprojective spaces.

Contribution

It introduces a method to approximate the Rees algebra via symmetric algebra resolutions, linking determinants to implicit equations under specific conditions.

Findings

Provides resolutions for symmetric algebra graded by divisor class group

Identifies a region in class group for choosing grading parameters

Offers geometric interpretation of extraneous factors in implicit equations

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.