Timeline for Existence of a holomorphic map between Riemann surfaces
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Aug 4, 2023 at 12:23 | vote | accept | Alexandre Eremenko | ||
| Aug 3, 2023 at 20:37 | comment | added | Sam Nead | Ok, even if it were true that all X have a hyperelliptic branched cover, we would not yet have a proof. The simple conectivity of $\mathbb{C}$ means it lifts to covers, not to branched covers. Sorry! | |
| Aug 3, 2023 at 18:36 | comment | added | user_1789 | Is the theorem of Picard (as stated by Nevanlinna) not stronger than stated above, in that it asserts that there are no nonconstant finitely punctured holomorphic planes in X, not just no nonconstant holomorphic planes? | |
| Aug 3, 2023 at 16:01 | comment | added | Dmitrii Korshunov | @SamNead if i'm not mistaken, wouldn't that mean that the corresponding finite extension $K/\mathbb{C}(t)$ of the field of rational functions embeds into a degree 2 extension $𝐿\supset 𝐾\supset \mathbb{C}(t)$ and then the Galois group $K/\mathbb{C}(t)$ should be a quotient of $\mathbb{Z}_2$ but the Galois group of a generic finite extension $K/\mathbb{C}(t)$ is $S_n$ or something. | |
| Aug 3, 2023 at 12:06 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
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| Aug 3, 2023 at 7:42 | comment | added | Sam Nead | Hmmm. Is every Riemann surface $X$ branched covered by a hyperelliptic surface? This would also give a proof (because the plane is simply connected). | |
| Aug 3, 2023 at 7:39 | comment | added | Sam Nead | I read Nevanlinna as saying "all of the ideas (for general $X$) are contained in the proof of the hyperelliptic case." | |
| Aug 3, 2023 at 7:36 | history | edited | Sam Nead | CC BY-SA 4.0 |
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| Aug 3, 2023 at 7:13 | answer | added | abx | timeline score: 5 | |
| Aug 2, 2023 at 22:24 | comment | added | paul garrett | @ChristianRemling, ah, ok, I'll delete my now-irrelevant comment... :) | |
| Aug 2, 2023 at 22:11 | comment | added | M.G. | @ChristianRemling: the operative word is "general", the identity map is not particularly general :-) | |
| Aug 2, 2023 at 21:19 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
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| Aug 2, 2023 at 20:49 | comment | added | Alexandre Eremenko | @abx: It is the last paragraph of Chap. 10, the proof of a theorem of Picard. He refers to Picard and Bloch on this, but none of them gives a clear statement, not speaking of a proof. | |
| Aug 2, 2023 at 20:21 | comment | added | Christian Remling | @abx: And what about the identity map $X\to X$ ? | |
| Aug 2, 2023 at 17:46 | comment | added | abx | This is not true: a general curve $X$ of genus $\geq 2$ has no nontrivial map to a curve of genus $\geq 1$. I couldn't find anything in Nevanlinna's book that resembles this statement. | |
| Aug 2, 2023 at 16:53 | history | asked | Alexandre Eremenko | CC BY-SA 4.0 |