Let $X:=\mathbb{P}^2\_K$ with $K$ algebraically closed field. Take $p\in X$ a point and $\mathcal{I}\_p$ its ideal sheaf. One can prove (using Serre Duality and the exact sequence defining $\mathcal{I}\_p$) that $Ext^1((\mathcal{I}\_p)\_p,\mathcal{O}\_{X,p})\cong K$, so there exists a non trivial extension \begin{equation\*}0\to\mathcal{O}\_X\to\mathcal{E}\to\mathcal{I}\_p\to 0
\end{equation\*}where $\mathcal{E}$ is a coherent sheaf over $X$. Now I would like to prove that $\mathcal{Ext}^i(\mathcal{E},\mathcal{O}\_X)\cong 0$ for all $i>0$. The case $i=1$ is clear since if $\mathcal{Ext}^1(\mathcal{E},\mathcal{O}\_X)$ were not be zero, then the connecting homomorphism from $\mathcal{Hom}(\mathcal{O}\_X,\mathcal{O}\_X)$ to $\mathcal{Ext}^1(\mathcal{I}\_p, \mathcal{O}\_X)$ would be surjective and this is not possible. So my first question



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> How can I prove that $\mathcal{Ext}^i(\mathcal{E},\mathcal{O}\_X)\cong 0$ for all $i\ge 2$?
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My first attempt was to take $\mathcal{Ext}^i(-,\mathcal{O}\_X)$ to the exact sequence above, but then I've realized that I should compute $\mathcal{Ext}^i(\mathcal{I}\_p,\mathcal{O}\_X)$ first. Here comes my second question



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> How can I compute $\mathcal{Ext}^i(\mathcal{I}\_p,\mathcal{O}\_X)$ for all $i\ge 0$?
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Computing Ext sheaves over complex projective plane
As stated above, from the LES in $\mathcal{E}xt$ we have
$$ \mathcal{E}xt^i(\mathcal{E}, \mathcal{O}\_X) \simeq \mathcal{E}xt^i(I\_p, \mathcal{O}\_X) $$
for all $i \geq 2$. How can we compute the term on the left hand side? The key is to observe the following fact:



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> A point in $\mathbf{P}^2$ is a complete intersection of two lines.
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What this means is that if $I\_p = \widetilde{I}$ for $I$ an ideal of $R := \mathbf{C}[x,y,z]$ generated by linear forms $f$ and $g$, we have a resolution (at the level of graded modules) given by
$$0 \to R(-2) \stackrel{c\mapsto (c,-c)}{\to} R(-1) \oplus R(-1) \stackrel{(a,b) \mapsto (af + bg)}{\to} I \to 0.$$
On the level of sheaves, this is
$$0 \to \mathcal{O}\_X(-2) \to \mathcal{O}\_X(-1) \oplus \mathcal{O}\_X(-1) \to I\_p \to 0.$$
It is now clear that $\mathcal{E}xt^i(I\_p, \mathcal{O}\_X) = 0$ for all $i \geq 2$, since in the LES it is sandwiched between sheafy exts of vector bundles (which are zero).


