Given a short exact sequence of coherent sheaves on a projective variety if we tensor the sequence with $\mathcal{O}(n)$ with a high enough $n$, everything becomes globally generated and by Serre's vanishing theorem taking the global sections becomes exact. I wonder whether it is possible to prove this without Serre's vanishing theorem. Serre's theorem applies for the proper varieties, I wonder whether the fact mentioned above remains true for let's say if we remove a high codimension subvariety from a projective variety? Like $\mathbb{P}^2\setminus \text{pt}$.
Serre’s vanishing theorem and global generation.
algebraic-geometry
Best Answer
Let $L$ be a line through $p\in\mathbb{P}^2$ and let $X=\mathbb{P}^2-\{p\}$ and $M=L-\{p\}$. You have an exact sequence of coherent sheaves, $0\to O_X(-1)\to O_X\to O_M\to 0$. Easy to see that $H^0(O_X(n))\to H^0(O_M(n))$ is not surjective for any $n$, since $H^0(O_M(n))$ is not finite dimensional over the base field.