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 $\mathbb{P}^1_{s,t}$.

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 $$ (tx_0 - sx_1, x_2x_4 - x_3^2, t^2x_2 - s^2x_4) $$