= Globally generated sheaf
{title2=$\Gamma(X,\mathcal F)\otimes\mathcal O_X\twoheadrightarrow\mathcal F$}
A sheaf of modules is globally generated when the evaluation map from its <global sections> is surjective. Equivalently, the germs of global sections span each stalk as a module over the local ring. For a <coherent sheaf> on a <projective variety>, every sufficiently positive twist is globally generated: embed the variety in projective space, present its direct-image sheaf as a quotient of a finite sum of $\mathcal O(-a)$, twist until those line bundles are globally generated, and restrict their generating sections to the variety.
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