# Topology of the product of projective varieties

I can't prove that the closed sets of a product of projective varieties is a zero locus of multihomogeneous polynomials. I'm taking the abstract point of view to the construction of product of varieties, i.e., if $$X$$,$$Y$$ are algebraic varieties (ringed spaces locally isomorphic to affine varieties) then we have affine open covers $$\{U_i\}, \{V_i\}$$ of $$X$$ and $$Y$$ respectively, and then we define the topology over $$X\times Y$$ saying that a subset $$C\subset X\times Y$$ is closed iff $$C\cap(U_i\times V_j)$$ is closed in $$U_i\times V_j$$ for all $$i,j$$. Then, my question is: if I have the product $$\mathbb{P} ^n\times\mathbb{P}^m$$ with the topology induced in the previous form, how I can prove that closed sets of this product are given by zeroes of bihomogeneous polynomials? Similarly, how I can prove that closed sets of $$\mathbb{P} ^n\times\mathbb{A}^m$$ are the zeroes of homogeneous polynomials in the first n variables?

• Does the Segre embedding $\mathbb P^n \times \mathbb P^m \hookrightarrow \mathbb P^{nm+n+m}$ help in any way? Commented Mar 1, 2023 at 17:30
• I understand the construction by the Segre embedding, we define a bijection between the product and a closed of $\mathbb{P}^{(n+1)(m+1)-1}$ and then we can transport the whole variety structure to this product, however what I want is to deduce this characterization of the closed sets from the construction that I have detailed above, since this is more general and thus I can understand other closed ones such as those of $\mathbb{P}^n\times\mathbb{A}^m$. Commented Mar 1, 2023 at 19:23

I believe red_trumpet's suggestion in their comment should answer your question: the Segre embedding $$i : \mathbb{P}^n \times \mathbb{P}^m \hookrightarrow \mathbb{P}^{nm + n + m}$$ defines a closed immersion of varieties; this implies that all closed subsets of $$\mathbb{P}^n \times \mathbb{P}^m$$ arise as the pull back of a closed subset of the projective space $$\mathbb{P}^{nm+n+m}$$. Any closed subset $$Z \subseteq \mathbb{P}^{nm+n+m}$$ is of the form $$V(s_1,\dots,s_r) = Z \subseteq \mathbb{P}^{nm+m+n}$$ where $$s_i \in \Gamma(\mathbb{P}^{nm+m+n},\mathcal{O}(d_i))$$ are global sections. The intersection of $$Z$$ with $$\mathbb{P}^n \times \mathbb{P}^m$$ will be given by $$V(i^\ast s_1,\dots,i^\ast s_r) = i^\ast Z \subseteq \mathbb{P}^n \times \mathbb{P}^m$$ where $$i^\ast s_j \in \Gamma(\mathbb{P}^n \times \mathbb{P}^m, i^\ast \mathcal{O}(d_i))$$ are the induced global sections. To conclude I think all you have to do is recognise the pullback bundles $$i^\ast \mathcal{O}(d_i)$$ on the product $$\mathbb{P}^n \times \mathbb{P}^m$$ as box-products of line bundles on $$\mathbb{P}^n$$ and $$\mathbb{P}^m$$, since their global sections are the polynomials you describe.
I'll describe this in an easy example where the numbers are small; I think the general argument should be clear: denote by $$[x_0:x_1]$$ and $$[y_0:y_1]$$ the projective coordinates on two copies of $$\mathbb{P}^1$$ and $$[z_0:z_1:z_2:z_3]$$ the coordinates on $$\mathbb{P}^3$$; the Segre embedding $$i : \mathbb{P}^1 \times \mathbb{P}^1 \hookrightarrow \mathbb{P}^3$$ is defined on the open patches $$\text{Spec}\left(\mathbb{Z}\left[\frac{x_1}{x_0},\frac{y_1}{y_0}\right]\right) \subseteq \mathbb{P}^1\times\mathbb{P}^1$$ $$\text{Spec}\left(\mathbb{Z}\left[\frac{z_1}{z_0},\frac{z_2}{z_0},\frac{z_3}{z_0}\right]\right) \subseteq \mathbb{P}^3$$ by the ring homomorphism given by $$\frac{z_1}{z_0} \mapsto \frac{x_1}{x_0}$$ $$\frac{z_2}{z_0} \mapsto \frac{x_1y_1}{x_0y_0}$$ $$\frac{z_3}{z_0} \mapsto \frac{y_1}{y_0}$$ and likewise on the other standard affine opens. The gluing data for the line bundle $$\mathcal{O}(1) \in \text{Pic}(\mathbb{P}^3)$$ on the intersection of standard opens $$\text{Spec}\left(\mathbb{Z}\left[\left(\frac{z_1}{z_0}\right)^{\pm 1},\frac{z_2}{z_0},\frac{z_3}{z_0}\right]\right) \subseteq \mathbb{P}^3$$ is given by multiplication by the invertible section $$\frac{z_0}{z_1} \implies$$ the induced gluing data for $$i^\ast \mathcal{O}(1)$$ on the open patch $$\text{Spec}\left(\mathbb{Z}\left[\left(\frac{x_0}{x_1}\right)^{\pm 1}, \frac{y_0}{y_1}\right]\right) \subseteq \mathbb{P}^1 \times \mathbb{P}^1$$ is given by multiplication by $$\frac{x_0}{x_1}$$ and similarly for the other intersections. If you stare at the formulas long enough you should be able to see that these define the gluing data for $$\mathcal{O}(1)\boxtimes \mathcal{O}(1) := \pi_1^\ast \mathcal{O}(1) \otimes \pi_2^\ast\mathcal{O}(1)$$ where $$\pi_1,\pi_2 : \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^1$$ are the two projections.
If you want to argue the similar result for the product $$\mathbb{P}^n \times \mathbb{A}^m$$ I think you can repeat the above argument by restricting the Segre embedding to this open subset of $$\mathbb{P}^n \times \mathbb{P}^m$$, since this will still define an immersion of varieties.