Uniform high-degree vanishing on a smooth projective curve ($\mathrm{deg}D\gg 0⟹H^{i>0}(C,\mathcal{F}(D))=0$)

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 $h^0(\mathcal{O}(A))\ge \mathrm{deg}A+1-g$. Every divisor of sufficiently large degree is therefore linearly equivalent to $n_{0}P+E$ 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.