Implicitization of rational hypersurfaces by syzygies with respect to coefficient ideals

Abstract

We study rational hypersurfaces $\mathcal{S}$ defined as the closure of the image of a generically finite rational map $\varphi :\mathcal{X}\to {\mathbb{P}}^{n+1}$, where $\mathcal{X}$ is an $n$-dimensional toric variety. We provide matrix representations for the implicitization of $\mathcal{S}$ that are constructed from the coefficients of linear syzygies and quadratic syzygies of the parametric equations. A central feature of our construction is the restriction of all coefficients in the Cox ring $R$ to a specific coefficient ideal $J$. In the two-dimensional case, this approach eliminates the need for $\varphi$ to be locally a complete intersection at the base points, that is, the determinant of the implicitization matrix is equal to a power of the implicit equation for arbitrary base points. This result generalizes several previous results in surface implicitization.