= Uniform high-degree vanishing on a smooth projective curve
{title2=$\deg D\gg0\implies H^{i>0}(C,\mathcal F(D))=0$}
For a fixed <coherent sheaf> on a <smooth projective curve>, one degree bound works for all <divisors on an algebraic curve>, rather than just powers of one ample divisor. Choose a point $P$. The complement of $P$ is affine, so the surjective transition maps on the finite-dimensional groups $H^1(C,\mathcal F(nP))$ eventually kill every class. The <Euler characteristic of a coherent sheaf> gives $h^0(\mathcal O(A))\ge\deg A+1-g$. Every divisor of sufficiently large degree is therefore linearly equivalent to $n_0P+E$ with $E$ effective. The point-supported quotient for adding $E$ transfers vanishing to that divisor. Torsion is first removed so that the relevant inclusions are injections of <locally free sheaves>.
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