I'm trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal slope) but it does not admit subbundles with greater slope. This is the simplest example I have in mind in order to explain why in dimension greater than 1 (of the Kahler manifold) one has to deal with subsheaves and not only subbundles when checking the stability of a vector bundle. I have problems in many part of the exposition which goes like this:

Let $p\in\mathbb{CP}^2$ be a point and $\mathcal{I}_p$ its ideal sheaf. Then, by using the Koszul resolution, we have the following short exact sequence of coherent sheaves $0\rightarrow\mathcal{O}(-2)\overset{f}{\rightarrow}\mathcal{O}(-1)\oplus\mathcal{O}(-1)\rightarrow\mathcal{I}_p\rightarrow 0$. Now take any non trivial sheaf homomorphism $\mathcal{O}(-2)\overset{g}{\rightarrow}\mathcal{O}$, which exists since it can be seen as a holomorphic section of $\mathcal{O}(-2)^*\otimes\mathcal{O}\cong\mathcal{O}(2)$, and let $\mathcal{E}$ be the push-out of $f$ and $g$ in the catgeory of coherent sheaf.

**Question 1:** How can we prove that $\mathcal{E}$ fits in the following sequence of sheaves $0\rightarrow\mathcal{O}\rightarrow\mathcal{E}\rightarrow\mathcal{I}_p\rightarrow 0$ ? Why is it exact?

*I have only understood that the first arrow is injective since $f$ it is.*

By using this exact sequence we know that $\mu(\mathcal{E}):=\frac{deg(\mathcal{E})}{rk(\mathcal{E})}=\frac{0}{2}=0$ but since $\mu(\mathcal{O})=0$ we conclude that $\mathcal{E}$ is not stable.

**Question 2:** Why $\mathcal{E}$ does not admit subbundles of greater slope?

**EDIT** *As Libli has wonderfully explained, this example shows that stability should be checked also on quotient sheaves and not only on quotient bundles. Moreover one has to prove that such a push-out $\mathcal{E}$ is indeed locally-free and not only coherent (as Libli has done).*

**EDIT 2** *If you are interested in this example you can find it in the wonderful book of Huybrechts-Lehn "The Geometry of Moduli Spaces of Sheaves" (Thm 5.1.1 and Ex. 5.1.2)*

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