Ideal sheaf of intersection

Question

Let $p_1,p_2,p_3$ be the points in the general position in $\mathbb{P}^2$ and $\mathcal I$ be their ideal sheaf. I want to find locally free resolution of $\mathcal I$. I can write $0\to\mathcal{O}(-2)\to\mathcal{O}(-1)^2 \to \mathcal I_{p_i}\to0$ and i know that $\mathcal I = \mathrm{im}\left(\mathcal I_{p_1}\otimes \mathcal I_{p_2}\otimes \mathcal I_{p_3} \to \mathcal O\right)$ but i has stuck with computing latter expression. Really i am trying to express $\mathcal{O}(1)$ on the blow-up of $\mathbb{P}^2$ in those three points in nice terms. And that resolution is for inclusion of that blow-up in projective bundle. Am i on the right way?

Answer 1

There is a resolution $$ 0 \to O(-3)^{\oplus 2} \to O(-2)^{\oplus 3} \to I_{p_1,p_2,p_3} \to 0. $$ For instance, if the points are $(1:0:0)$, $(0:1:0)$, and $(0:0:1)$, then the morphism is given by $$ \left(\begin{matrix} x & y & 0 \\ 0 & y & z \end{matrix}\right). $$