Equations Defining Certain Graphs

Abstract

Consider the rational map $\varphi :{\mathbb{P}}_{\mathbf{k}}^{n-1}\frac{[f_0:\cdots :f_n]}{⇢}{\mathbb{P}}_{\mathbf{k}}^n$ defined by homogeneous polynomials $f_0,\dots ,f_n$ of the same degree $d$ in a polynomial ring $R=\mathbf{k}[x_1,\dots ,x_n]$ over a field $\mathbf{k}$. Suppose that $I=(f_0,\dots ,f_n)$ is a height two perfect ideal satisfying $\mu \left(I_p\right)\le dim{R}_p$ for $p\in Spec\left(R\right)\setminus V(x_1,\dots ,x_n)$. We study the equations defining the graph of $\varphi$ whose coordinate ring is the Rees algebra $R\left[\mathit{It}\right]$. We provide new methods to construct these equations using the work of Buchsbaum and Eisenbud. Furthermore, for certain classes of ideals satisfying the conditions above, our methods lead to explicit equations defining Rees algebras of the ideals in these classes. These classes of examples are interesting in that there are no known methods to compute the defining ideal of the Rees algebra of such ideals. Our new methods also give effective criteria to check that $\varphi$ is birational onto its image.