= Uniform Serre vanishing for two ample twists
{title2=$H^{i>0}(X,\mathcal F\otimes L^m\otimes B^t)=0$}
For <ample> <line bundles> $L,B$ and a <coherent sheaf> $\mathcal F$ on a <projective scheme>, one $M$ gives $H^i(X,\mathcal F\otimes L^m\otimes B^t)=0$ for every $i>0$, $m\ge M$ and $t\ge0$. If $B$ is <very ample>, <Serre vanishing> for the finitely many fixed twists $\mathcal F(-iB)$ makes $\mathcal F\otimes L^m$ zero-regular. Persistence of <Castelnuovo–Mumford regularity> gives all nonnegative $B$-twists. For general <ample> $B$, use a <very ample> power and treat finitely many residues of $t$.
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