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I am trying to read Friedman's "Algebraic surfaces and holomorphic vector bundles". I am unable to follow a claim (on pg 32) that any globally generated rank 2 vector bundle (say) $E$ on a complex algebraic surface $X$ has $\mathcal{O}_X$ as a sub bundle. My doubt is this that if $s \in Hom(\mathcal{O}_X, E)$ be a global section, then will the map $\mathcal{O}_X \rightarrow E$ be an injection on the vanishing locus of $s$? |
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If $s\ne 0$, then the map $\mathcal O_X\to E$ induced by $s$ is injective. Indeed, suppose for some non-empty open subset $U$ of $X$, the map $$O_X(U)\to E(U), \quad f\mapsto fs|_U $$ is not injective. Then there exists $f\in O_X(U)$ non-zero such that $fs|_U=0$. As $E$ is locally free, this implies that $s|_U=0$. But again because $E$ is locally free and $X$ is integral, the restriction map $E(X)\to E(U)$ is injective, so $s|_U\ne 0$. Contradiction. I think you make confusion between the vanishing locus of $s$ and its zero divisor. |
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