Timeline for A purely algebraic argument for existence of a section of a smooth projective morphism to the projective line
Current License: CC BY-SA 4.0
5 events
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| Jul 15, 2019 at 8:29 | comment | added | Libli | Since any smooth morphism admits a section locally in the étale topology, I am wondering if one can not prove this using: i) the simply connectedness of $\mathbb{A}^1$ ii) the fact that any rational map between projective curves can be extended to an actual morphism? | |
| Jul 12, 2019 at 16:10 | comment | added | Ariyan Javanpeykar | It is not so clear to me what one should expect when $k=\overline{\mathbb{F}_p}$. | |
| Jul 12, 2019 at 15:59 | comment | added | Ariyan Javanpeykar | Now, use "spreading out" (of your section) over a variety over $k$, and then specialize the spread-out section to get a section of $f$. | |
| Jul 12, 2019 at 15:58 | comment | added | Ariyan Javanpeykar | If $k$ is an algebraically closed field of characteristic zero and $f:X\to \mathbb{P}^1$ is a smooth projective morphism then $f$ will have a section. You can prove this by "reducing" to the case that $k=\mathbb{C}$. This is an example of the so-called "Lefschetz principle". To get a feeling of how this goes, assume for simplicity that $k\subset \mathbb{C}$. Now, consider a section $\sigma:\mathbb{P}^1_{\mathbb{C}}\to X_{\mathbb{C}}$ of $f_{\mathbb{C}}$. | |
| Jul 12, 2019 at 8:20 | history | asked | user142965 | CC BY-SA 4.0 |