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Let $\mathcal{I}_p$ be the ideal sheaf of a point $p\in\mathbb{P}^2$, and let $\mathcal{E}$ be the rank two locally free sheaf over $\mathbb{P}^2$ determined by the following exact sequence: $$0\rightarrow \mathcal{O}_{\mathbb{P}^2}\rightarrow \mathcal{E}\rightarrow \mathcal{I}_{p}\rightarrow 0.$$ How could one compute the Chern classes of $\mathcal{E}$ ?

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Of course $c(\mathcal E) = c(I_p)$. Computing the Chern class of $I_p$ can be done with the sequence $$ 0 \to I_p \to O_{\Bbb P^2} \to O_p \to 0 $$ which is done here. You obtain $c_1(I_p) = 0$ and $c_2(I_p) = 1$.

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