How do I find the algebra representing the projective bundle of a direct sum of line bundles over a projective space?

Question

I am trying to learn how to compute the projective bundle $\mathbb{P}(\mathcal{O}(a_1)\oplus \cdots \mathcal{O}(a_k))$ over some projective space using relative proj. How can I find a presentation for the ideal $I$ giving the closed subscheme of $\mathbb{P}^n\times\mathbb{P}^m$? For example, I want to understand how I can find an algebra representing the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus \mathcal{O}(1))$ over $\mathbb{P}^1_{s,t}$.

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For my example, I have to twist by $\mathcal{O}(1)$ to get a vanishing $h^1$. Then, there is a map $$ \mathcal{O}^{\oplus 5} \xrightarrow{ \begin{bmatrix} s \\ t \\ s^2 \\ st \\ t^2 \end{bmatrix}} \mathcal{O}(1)\oplus\mathcal{O}(2) \to 0 $$ All that's left is finding the generators of the ideal $I \subseteq \mathbb{C}[s,t][x_0,\ldots, x_4]$, which I think has the presentation $$ (tx_0 - sx_1, x_2x_4 - x_3^2, t^2x_2 - s^2x_4) $$

Answer 1

OK, I guess I will write it here as it might need more room.

You know that $\mathbb P(\mathscr E)$ remains the same if you twist it by a line bundle. So, choose a sufficiently ample line bundle $\mathscr L$ such that $\mathscr E\otimes \mathscr L$ is generated by global sections and switch $\mathscr E$ with $\mathscr E\otimes \mathscr L$. In other words, you may assume that $\mathscr E$ is generated by global sections.

Now you have a surjective morphism $$ \mathscr O^{\oplus (m+1)}_X \to \mathscr E. $$ which using the properties of projective bundles gives you an embedding $$ \mathbb P(\mathscr E) \to X\times \mathbb P^m. $$

Now, you can probably write down an absolute Proj for $\mathbb P^2\times \mathbb P^m$ and then your original projective bundle is a closed subscheme in there, so you just have to figure out its ideal.

Ta-da.