Fix $n$ and let $0\leftarrow \mathcal{F}\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(a_i)\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(b_i)\leftarrow \cdots$ be an exact sequence.
Then we can say that $\mathbb{P}(\mathcal{F})\hookrightarrow \mathbb{P}(\bigoplus \mathcal{O}_{\mathbb{P}^n}(a_i))$ but what more can we say about $\mathbb{P}(\mathcal{F})$?
Is there a way to write the ring of $\mathbb{P}(\mathcal{F})$ and the equations given by the map $\bigoplus \mathcal{O}_{\mathbb{P}^n}(a_i)\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(b_i)$?