Let \(\mathcal F\) be the structure sheaf of \(\CP^2\), and let \(\mathcal I\) be the sheaf of ideals defining two distinct reduced points \(p,q\in\CP^2\). Define a presheaf \(\mathcal G\) by \[ \mathcal G(U)=\mathcal F(U)/\mathcal I(U) \] for every Zariski-open set \(U\subseteq\CP^2\).
Show that \(\mathcal G\) is not a sheaf.
Determine the associated sheaf \(\mathcal G^+\); that is, determine \(\mathcal G^+(U)\) for every Zariski-open set \(U\subseteq\CP^2\).