How do we construct a product of projective spaces as a sequence of projective bundles? Sometimes, it is useful to treat a variety as a projective bundle over a projective space. Therefore, I would like to understand several examples below.
For example, the product variety $\mathbf{P}^1\times \mathbf{P}^2$ is isomorphic to the projective bundle
$$\mathbf{P}(\mathcal{O}(-a)\oplus \mathcal{O}(-a) )\to \mathbf{P}^2.$$
The product variety $\mathbf{P}^2\times \mathbf{P}^2$ is isomorphic to the projective bundle
$$\mathbf{P}(\mathcal{O}(-b)\oplus \mathcal{O}(-b)\oplus \mathcal{O}(-b) )\to \mathbf{P}^2.$$
I would like to know why those varieties are isomorphic (rigorous proof and how to see this intuitively).
Moreover, I would like to know if this is true in general, that is,
for $\mathbf{P}^{n_1}\times\cdots \times \mathbf{P}^{n_r}$, is there a projective bundle structure like the above examples?
Any references and answers are welcome!
 A: Question: For example, the product variety $\mathbf{P}^1\times \mathbf{P}^2$ is isomorphic to the projective bundle
$$\mathbf{P}(\mathcal{O}(-a)\oplus \mathcal{O}(-a) )\to \mathbf{P}^2.$$
The product variety $\mathbf{P}^2\times \mathbf{P}^2$ is isomorphic to the projective bundle
$$\mathbf{P}(\mathcal{O}(-b)\oplus \mathcal{O}(-b)\oplus \mathcal{O}(-b) )\to \mathbf{P}^2.$$
I would like to know why those varieties are isomorphic (rigorous proof and how to see this intuitively).
Answer: Let $C$ be the projective line and $S$ the projective plane and let $L(d):=\mathcal{O}_C(d), M(d):=\mathcal{O}_S(d)$.
In the first case you get isomorphisms
$$ \mathbb{P}((L(-a)\oplus L(-a))^*) \cong \mathbb{P}((L(-a)\otimes(\mathcal{O}_S^2))^*) \cong \mathbb{P}((\mathcal{O}_S^2)^*) \cong S \times C.$$
This is because for any locally trivial finite rank sheaf $E$ it follows
$$\mathbb{P}((L(d)\otimes E)^*) \cong \mathbb{P}(E^*)$$
and the fact that for a trivial vector bundle $E:=\mathcal{O}_S^{r+1}$ it follows
$$Proj(E^*) \cong \mathbb{P}^r_S \cong \mathbb{P}^r \times \mathbb{P}^2.$$
When you twist $E$ with an invertible sheaf $L(d)$, you get isomorphisms (see Hartshorne Ex.II.7.10). A similar proof shows the second case.
Question: Moreover, I would like to know if this is true in general, that is, for $\mathbf{P}^{n_1}\times\cdots \times \mathbf{P}^{n_r}$, is there a projective bundle structure like the above examples?
Any references and answers are welcome!"
Answer: Use induction as follows: Let $Y:=\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_l}:=X\times \mathbb{P}^{n_l}$. Let $E_l:=\mathcal{O}_X^{n_l+1}$. It follows
$$\mathbb{P}(E_l^*) \cong X \times \mathbb{P}^{n_l} \cong Y.$$
Hence you can construct any finite product $Y:=\prod \mathbb{P}^{n_i}$ as a sequence of projective bundles.
Example: If $l=3$ and $Y:=\mathbb{P}^{n_1} \times \mathbb{P}^{n_2}\times \mathbb{P}^{n_3}$ and if you fix the projection map
$$\pi_1: Y \rightarrow \mathbb{P}^{n_1}$$
with fiber $F:=\mathbb{P}^{n_2}\times \mathbb{P}^{n_3}$, the fiber is trivially not isomorphic to projective space. There is by the above argument a sequence of maps
$$Y\cong \mathbb{P}(E_2^*) \rightarrow \mathbb{P}(E_1^*) \rightarrow \mathbb{P}^{n_1}$$
with $E_i$ a locally trivial sheaf for $i=1,2$.
Application: You may use the "projective bundle formula" and an induction to calculate the Chow ring (or Grothendieck ring) of any finite product of projective spaces: The formula says
$$CH^*(\mathbb{P}(E^*)) \cong CH^*(X)[t]/(t^{e})$$
where $e:=rk(E)$ (Hartshorne, CH.AppendixA.3).
