Let $\mathcal{E}$ be a rank two vector bundle on $\mathbb{P}^2$ fitting in the following exact sequence


$$0\rightarrow \mathcal{O}\_{\mathbb{P}^2}\rightarrow \mathcal{E}\rightarrow \mathcal{I}\_p(-1)\rightarrow 0$$


where $\mathcal{I}\_p$ is the ideal sheaf of a point $p\in\mathbb{P}^2$. Assume that $\mathcal{E}$ is normalized, $c\_{1}(\mathcal{E})\in\{-1,0\}$.


How can we compute the Chern classes of $\mathcal{E}$ ?



Chern classes of a vector bundle
As $\mathcal{O}\_{\mathbb{P}^2}$ is trivial, then multiplicativity of Chern classes in exact sequences implies:
$$
c\_\*(\mathcal{E}) = c\_\*(\mathcal{I}\_p(-1)).
$$
We can compute $c\_\*(\mathcal{I}\_p(-1))$ by noting that $p\in \mathbb{P}^2$ is a complete intersection of 2 lines. Therefore, the $(-1)$-twist of the Koszul complex is exact:
$$
0\rightarrow \mathcal{O}\_{\mathbb{P}^2} (-3) \rightarrow \mathcal{O}\_{\mathbb{P}^2}(-2)^{\oplus 2} \rightarrow \mathcal{I}\_p(-1)\rightarrow 0.
$$
Thus
$$
c\_\*(I\_p(-1)) = c\_\*(\mathcal{O}\_{\mathbb{P}^2}(-2)^{\oplus 2})/c\_\*(\mathcal{O}\_{\mathbb{P}^2}(-3)) = (1-4h+4h^2)/(1-3h)\\
=(1-4h+4h^2)(1+3h+9h^2)\\
= 1-h+(9-12+4)h^2 = 1-h+h^2,
$$
where $h\in H^2(\mathbb{P}^2,\mathbb{Z})$ is the hyperplane generator. Thus: $c\_1(\mathcal{E})=-1$ and $c\_2(\mathcal{E}) = 1$.


I.e. $\int\_{[\mathbb{P}^2]} c\_2(\mathcal{E}) = 1$.


One can also use Grothendieck-Riemann-Roch to compute $c\_\*(\mathcal{O}\_p) = 1-h^2$, and then use the ideal sequence for $\mathcal{O}\_p$.


