Timeline for Computing the first sheaf cohomology
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 18, 2023 at 15:03 | comment | added | Leo D | @abx Thanks again. This is also a great suggestion. I will wait and see that if there is any other method/trick to compute $h^1$. | |
| Jul 18, 2023 at 6:34 | comment | added | abx | There are other ways to compute $h^1$. For instance, suppose $X$ is regular ($h^1(\mathscr{O}_X)=0$), and $D$ is a disjoint union of $r$ integral curves. Then the exact sequence $$0\rightarrow \mathscr{O}_X(-D)\rightarrow \mathscr{O}_X\rightarrow \mathscr{O}_D\rightarrow 0$$ gives $h^1(-D)=r-1$. | |
| Jul 18, 2023 at 5:27 | comment | added | Leo D | @abx Thanks for this nice example. So can I ask if every example you know comes from a similar method? That is, we compute both $h^0(F)$ and $h^2(F)$, and then use Riemann-Roch. | |
| Jul 17, 2023 at 18:22 | comment | added | abx | There are so many examples that it is hard to choose. Here is one: suppose your surface admits an elliptic fibration $X\rightarrow \mathbb{P}^1$, and let $F$ be a fiber. Then $\mathscr{O}(F)=p^*\mathscr{O}_{\mathbb{P}^1}(1)$, from which one gets $h^0(nF)=n+1$. We have $h^0(K-nF)=0$ for $n$ large, hence $h^1(nF)= n+1-\chi (\mathscr{O}_X)$ by Riemann-Roch. | |
| Jul 17, 2023 at 16:54 | comment | added | Leo D | Hi @LSpice, exactly! I am looking for nontrivial examples. Thanks for helping me to clarify my question. | |
| Jul 17, 2023 at 16:29 | history | edited | LSpice | CC BY-SA 4.0 |
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| Jul 17, 2023 at 16:28 | comment | added | LSpice | By "beside of different kinds of vanishing theorem", do you mean just that you are looking for examples where the cohomology is non-trivial? | |
| Jul 17, 2023 at 16:18 | history | asked | Leo D | CC BY-SA 4.0 |