Timeline for Do Deligne-Mumford curves also have rational functions
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 11, 2013 at 15:53 | vote | accept | Keesjan | ||
| Sep 11, 2013 at 15:53 | comment | added | Keesjan | Thnx for this. My silly question was aiming at the statement that "if $\mathcal X$ has stacky points, then no morphism $\mathcal X\to \mathbf P^1$ is representable". Thank you very much for your answer. :) | |
| Sep 11, 2013 at 15:37 | comment | added | Jason Starr | @Keesjan: You know what you were asking about better than I do. However, my interpretation of a "finite morphism" of stacks is a morphism that is representable by finite morphisms. If $\mathcal{X}$ has stacky points, then no morphism $\mathcal{X}\to \mathbb{P}^1$ is representable. The construction above produces a representable morphism $\mathcal{X}\to \mathbb{P}(1,n)$. | |
| Sep 11, 2013 at 13:00 | comment | added | Keesjan | My apologies for the unusual etiquette. My follow-up question is actually related to your answer. In fact, you construct a finite morphism $\mathcal X\to X \mathbf P^1$ which factors through some $\mathbf P^1(n,m)$. So this is probably very stupid, but why bother finding a factorization? Doesn't the morphism $\mathcal X\to X \mathbf P^1$ already answer the question? Is it not a "rational function"? Hope I'm making sense... | |
| Sep 11, 2013 at 12:20 | comment | added | Jason Starr | @Keesjan: Usually it is best to ask a follow-up question as a separate question on MO. I am not sure I understand your follow-up question: if I take $n=m=1$, then the morphism $v:\mathbb{P}(n,m)\to \mathbb{P}^1$ is an isomorphism. | |
| Sep 11, 2013 at 8:26 | comment | added | Keesjan | Hi, is it 'easy' to write down an example of a morphism $\mathcal X \to X \to \mathbf P^1$ which does not factor through a 1-morphism $F:\mathcal X \to \mathbf P^1(n,m)$ for any $n,m$? | |
| S Sep 10, 2013 at 15:34 | history | answered | Jason Starr | CC BY-SA 3.0 | |
| S Sep 10, 2013 at 15:34 | history | made wiki | Post Made Community Wiki by Jason Starr |