Embeddings into products of projective spaces and multi graded rings Let X be a variety over a field $k$. Assume that it is embedded into product of two projective spaces
$$
i : X \subset \mathbb{P}^n \times \mathbb{P}^m.
$$
I want to construct a graded algebra that corresponds to this embedding in the same spirit as we construct homogeneous coordinate ring of a variety embedded into a single projective space. I guess, in this case I need two line bundles 
$$
\mathcal{L} = i^* \mathcal{O}(1,0),
$$
and
$$
\mathcal{M} = i^* \mathcal{O}(0,1),
$$
and corresponding algebra 
$$
R = \bigoplus_{i \geq 0, j \geq 0} H^0(X, \mathcal{L}^i \otimes \mathcal{M}^j)
$$
is bigraded. 
I think something like this should be well known. Where can I find discussion of this construction and related questions?
 A: $\def\P{{\mathbb P}}\def\O{{\mathcal O}}$Yes you are exactly right. One way to describe a closed subvariety of $\P^n \times \P^m$ is to give it as the vanishing of some homogeneous ideal $I$ of the bigraded algebra $A = k[x_0, \ldots, x_n, y_0, \ldots, y_m]$ where a polynomial is homogeneous if it is so in the $x_i$ and $y_j$ separately. In this case you get a bigraded coordinate ring $A/I$. Then $A/I$ is exactly your $R$. 
Let us call $\O(k,l) = \pi_1^*\O(k) \otimes \pi_2^*\O(l)$ where $\pi_i$ are the projections to the first and second projective spaces respectively. Then the sections of $\O(k,l)$ on $\P^n \times \P^m$ are the polynomials in $A$ that are homogeneous degree $k$ in the $x_i$ and homogeneous degree $l$ in the $y_j$ and so 
$$
A = \bigoplus_{k,l \geq 0} H^0(\P^n \times \P^m, \O(k,l))
$$
and 
$$
A/I = \bigoplus_{k,l \geq 0} H^0(X, i^*\O(k,l))
$$
which is exactly your $R$ above. 
However, this is not the usual projective coordinate ring of a projective variety, i.e., you can't take Proj of this to obtain $X$. The way to view this as a coordinate ring is through a generalization of Proj, sometimes called multi-Proj. 
$\P^n \times \P^m$ is the GIT quotient $\operatorname{Spec} A//(\mathbb{G}_m \times \mathbb{G}_m) = (\mathbb{A}^{n+1} \times \mathbb{A}^{m+1})//(\mathbb{G}_m \times \mathbb{G}_m)$ where each multiplicative groups act on the corresponding factor of the product independently. Then $X$ can be given as the GIT quotient $(\operatorname{Spec}A/I)//(\mathbb{G}_m \times \mathbb{G}_m)$ where $\operatorname{Spec}A/I$ inherits a $\mathbb{G}_m \times \mathbb{G}_m$ action from the fact that $I$ was bigraded. This generalizes the usual Proj construction which is the GIT quotient of a singly graded coordinate ring by $\mathbb{G}_m$. 
For more see these MathOverflow questions: https://mathoverflow.net/questions/48147/generalize-the-proj-construction and https://mathoverflow.net/questions/126722/why-does-the-naive-choice-of-homogeneous-coordinate-ring-of-a-product-of-project. 
