Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Equations defining certain graphs   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.

17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con