Let $X$ and $Y$ be projective varieties (both in $\mathbb P^n$) and let $\phi:\mathbb P^n\times \mathbb P^n\to\mathbb P^m$ be a rational map defined by homogeneous equations $f_0,\dots,f_m$ where $\phi(x:y) = (f_0: \ldots :f_m)$. The graph of $\phi$ has points of the form $(x_0:\ldots:x_n:y_0:\ldots:y_n:f_0:\ldots:f_m)$.
I know from literature that the graph of $\phi$ is a projective variety (Harris), but I'm having trouble determining what the ideal defining this variety is. Intuitively I thought it may just be the ideals of $X$ and $Y$ along with the ideal defining the closure of $\phi$ in $P^m$, but I seem to be missing equations. It has been hinted at that this is not an easy question to answer in the projective case, but I have found an example done in the affine case.
My question is, can I find all of the equations in the projective case by looking at all the affine patches of the projective graph of $\phi$ (i.e. performing homogenization on the equations arising on each of the affine patches)?