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 $\bigoplus \mathcal{O}\_{\mathbb{P}^n}(a\_i)\leftarrow \bigoplus \mathcal{O}\_{\mathbb{P}^n}(b\_i)$?



Projective bundles
According to the long exact sequence, the sheaf $\mathcal{F}$ is coherent and it is over ${P}^n$, so there is a graded module $M$ such that the coherent sheaf $\tilde{M}$ is just the $\mathcal{F}$, the one may construct $\mathbb{P}(\mathcal{F})$ by using this $M$. The way to construct this $M$ can be found in GTM52 ALGEBRAIC GEOMETRY, in chapter2 section5.  

I hope this does a little help.


