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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 $\bigoplus \mathcal{O}_{\mathbb{P}^n}(a_i)\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(b_i)$?

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  • $\begingroup$ I suggest that you try to work this out for yourself, perhaps in the case that all $a_i$ and $b_j$ equal $0$. The basic feature of that special case extends to the general case. $\endgroup$ Oct 3, 2013 at 12:42

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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.

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