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    Sheafification of a quotient presheaf for two reduced points in $\CP^2$

    P-UNL970953-04-AG4

    Area
    Algebraic Geometry
    Seen at
    UNL
    Years
    2004
    Status
    draft
    Contents

    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 \(U\subseteq\CP^2\).

    1. Show that \(\mathcal G\) is not a sheaf.

    2. Determine the associated sheaf \(\mathcal G^+\); that is, determine \(\mathcal G^+(U)\) for every Zariski-open set \(U\subseteq\CP^2\).

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