= Polynomial bound for sections of a fixed divisor
{title2=$h^0(X,\mathcal F(mD))=O(m^d)$}
For a <coherent sheaf> $\mathcal F$ of support dimension $d$ on a <projective scheme> and a fixed <Cartier divisor> $D$, $h^0(X,\mathcal F(mD))=O(m^d)$. Choose a sufficiently high ample divisor $A$ whose section avoids the <associated points> of $\mathcal F$ and for which $D+A$ is ample. Multiplication by the $m$th power of that section injects $\mathcal F(mD)$ into $\mathcal F(m(D+A))$. The ample <Hilbert polynomial>, together with <Serre vanishing>, bounds the latter section space by $O(m^d)$.
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