Topology of the product of projective varieties

Question

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?

Answer 1

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.

I'm not sure this was exactly what you were looking for, and I apologise if my explanations were a little too obvious :p I hope I still managed to help somehow :)