Timeline for Finite generation of global sections of an invertible sheaf on a quasi-projective scheme
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 3, 2016 at 17:18 | vote | accept | A.G | ||
| Jun 1, 2016 at 11:16 | answer | added | Jason Starr | timeline score: 5 | |
| Jun 1, 2016 at 10:53 | comment | added | Jason Starr | I can see how to make a counterexample with $\mathcal{F}$ a pure, torsion-free sheaf of rank $1$, and $X$ is an integral, local complete intersection scheme (that is not normal). Namely, $X$ is a $\mathbb{P}^1$-bundle over a nodal plane cubic, and $\mathcal{F}$ is the pushforward of the structure sheaf of the normalization. However, I do not see an example with $\mathcal{F}$ an invertible sheaf. | |
| Jun 1, 2016 at 10:31 | comment | added | Lazzaro Campeotti | It must be contagious. Note that the version without "invertible" was asked and answered here: math.stackexchange.com/questions/1805890/… with the same counterexample. | |
| Jun 1, 2016 at 10:25 | comment | added | Jason Starr | I missed that hypothesis! | |
| Jun 1, 2016 at 10:17 | comment | added | Lazzaro Campeotti | @JasonStarr: the question asks for $F$ invertible. | |
| Jun 1, 2016 at 10:14 | comment | added | Jason Starr | Ben Lim's example is easily modified to a true counterexample. Let $X$ be $\mathbb{P}^2_k$, let $\mathcal{F}$ be the coherent sheaf that is the pushforward of the structure sheaf of a line $\mathbb{P}^1_k\subset \mathbb{P}^2_k$. Let $\infty\in \mathbb{P}^1$ be a $k$-point. Let $U$ be the open complement of that $k$-point inside $X$. Then $\Gamma(U,\mathcal{O}_X)$ is canonically $k$, but $\Gamma(U,\mathcal{F})$ is $k[t,t^{-1}]$ as a $k$-algebra, thus not a finite dimensional $k$-vector space. | |
| Jun 1, 2016 at 6:21 | comment | added | A.G | @Ben Lim: Isn't it even cyclic in that case? | |
| Jun 1, 2016 at 6:03 | review | First posts | |||
| Jun 1, 2016 at 6:57 | |||||
| Jun 1, 2016 at 6:01 | comment | added | A.G | Sorry if too elementary, but I had no answer in math.stackexchange.com/questions/1805972/… | |
| Jun 1, 2016 at 6:00 | history | asked | A.G | CC BY-SA 3.0 |