Timeline for Affinization map of cotangent bundle is proper/projective?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 8, 2023 at 15:40 | history | edited | Filip | CC BY-SA 4.0 |
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| Apr 15, 2023 at 15:27 | comment | added | Filip | @onefishtwofish Many thanks! now the comments above make sense. | |
| Apr 13, 2023 at 13:27 | comment | added | onefishtwofish | @Filip In algebraic geometry, the total space of a vector bundle E is defined as the relative Spec of $Sym(E)$ or $Sym(E^{\vee})$ depending on what universal property you want it to satisfy. Let's just take the first convention for simplicity. In the case you are looking at, $E=L$ is a line bundle and $Sym(L)$ will be a direct sum of $L^{\otimes r}$. So, if all of the global sections of $L^{\otimes r}, r>0$ vanish, you will just have constant functions on Tot E. | |
| Apr 9, 2023 at 19:10 | comment | added | Filip | Let us continue this discussion in chat. | |
| Apr 9, 2023 at 18:35 | comment | added | Filip | Maybe I am misunderstanding, but let me just say that global sections of the structure sheaf of $T^*X$ (seen as a variety) and global sections of $T^*X$ (seen as a sheaf over $X$) are two different things. That's why the variety $T^*\mathbb{P}^1$ has global sections, although the bundle $T^*\mathbb{P}^1=\mathcal{O}(-2)\rightarrow \mathbb{P}^1$ does not. | |
| Apr 9, 2023 at 18:05 | comment | added | Filip | I see. As I still don't understand your argument, I guess that there is some statement that you are using in it? Regarding the comparison between global sections of the structure sheaf of $T^*X$ and of $(T^*X)^{\otimes r},$ for a projective variety $X.$ | |
| Apr 9, 2023 at 14:50 | comment | added | Jason Starr | There are two opposite conventions on the meaning of that: the one in EGA and the other one. That is precisely why I gave one example for each convention. | |
| Apr 9, 2023 at 13:44 | comment | added | Filip | Well, the $T^*\mathbb{P}^1$ is not a counterexample actually, as its affinization is the well-known resolution $T^*\mathbb{P}^1 \rightarrow \mathbb{C}^2/\mathbb{Z}/2,$ which is projective. | |
| Apr 9, 2023 at 0:53 | comment | added | Jason Starr | All positive tensor powers of the cotangent bundle on the projective line have vanishing global sections. All positive tensor powers of the tangent bundle on hyperbolic curves have vanishing global sections. | |
| Apr 9, 2023 at 0:35 | comment | added | Filip | Hmm, I see. An example of this would be helpful? | |
| Apr 9, 2023 at 0:29 | comment | added | Jason Starr | No, that is not true. It can happen that the only global sections of the structure sheaf are constants. Then the morphism is constant, hence non-proper, since $Y$ is not proper. | |
| Apr 8, 2023 at 22:13 | history | asked | Filip | CC BY-SA 4.0 |