Let $X$ be a projective variety over a noetherian ring and let $L=\mathcal{O}_X(1)$ be a very ample line bundle. If $F$ is a coherent sheaf on $X$ Serre's vanishing theorem tells us that there is $n_0$ depending on $F$ s.t. for $i>0$ and $n \geq n_0$, $H^i(X,F(n))=0$.
Let $F=\mathcal{O}_X$, how to find $n_0$ that works in this case? Is it true that $n_0=1$? In other words a very ample line bundle does not have any higher cohomology.