= Euler-characteristic ampleness criterion
{c}
{title2=$\chi(V,\mathcal O_V(mD))\to+\infty$}
A <Cartier divisor> $D$ on a <projective scheme> is <ample> exactly when $\chi(V,\mathcal O_V(mD))\to+\infty$ for every positive-dimensional integral closed <subvariety> $V$, including irreducible components. For the converse induct on dimension: restrictions to hyperplane sections are <ample>, so <higher cohomology vanishing from an ample hyperplane restriction> gives $h^0=\chi+h^1\to\infty$. A nonzero section vanishing at a chosen point then exists. The <vanishing-section ampleness criterion> finishes. Divergence alone does not imply a positive top-degree coefficient.
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