Uniform Serre vanishing for two ample twists ($H^{i>0}(X,\mathcal{F}\otimes L^m\otimes B^t)=0$)

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 $t\ge 0$. 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 $t$.

Let $H$ be a very ample effective Cartier divisor on a projective scheme $X$, and suppose $L{|}_H$ is ample. Then $H^i(X,L^m)=0$ for $i\ge 2$ and sufficiently large $m$. By uniform Serre vanishing for two ample twists, cohomology of $L^m(tH){|}_H$ vanishes uniformly for $t\ge 0$. The restriction sequence identifies the higher groups of $L^m(tH)$ and $L^m((t+1)H)$ for $i\ge 2$. For fixed $m$, large $t$ kills them by Serre vanishing for $H$; descend to $t=0$. No $H^1$ vanishing is claimed.