Equations defining certain graphs

TL;DR

This paper develops new methods to explicitly determine the equations defining the Rees algebra of certain height two perfect ideals, which in turn describe the graphs of rational maps, and provides criteria for birationality.

Contribution

It introduces novel techniques based on Buchsbaum-Eisenbud work to construct defining equations of Rees algebras for specific classes of ideals, including explicit cases previously inaccessible.

Findings

New explicit equations for Rees algebras of certain ideals.

Effective criteria for birationality of the rational map.

Applicable to classes of ideals with no prior known methods.

Abstract

Consider the rational map $\phi: \mathbb{P}^{n-1}_{\mathbf k} \stackrel{[f_0:\cdots: f_n]}{\longrightarrow} \mathbb{P}^{n}_{\mathbf k}$ 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 $I=(f_0,\dots,f_n)$ is a height two perfect ideal satisfying $\mu(I_p)\leq\dim R_p$ for $p\in \operatorname{Spec} (R) \setminus V(x_1,\dots, x_n)$. We study the equations defining the graph of $\phi$ whose coordinate ring is the Rees algebra $R[It]$. We provide new methods to construct these equations using 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. These new methods also give rise to effective criteria to check that $\phi$ is birational onto its image.