Timeline for coarse moduli space $X(2)$
Current License: CC BY-SA 3.0
20 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Oct 19, 2016 at 22:11 | comment | added | Adel BETINA | I want to understand the geometry of the $p$-adic rigid curve associated to $X(\Gamma_0(p) \cap \Gamma(N)$ and its formal model. | |
| Oct 19, 2016 at 21:53 | comment | added | Will Chen | @AdelBETINA If you don't mind, may I ask what you're working on? | |
| Oct 19, 2016 at 21:51 | vote | accept | Adel BETINA | ||
| Oct 19, 2016 at 21:51 | comment | added | Will Chen | @AdelBETINA It's obviously connected. It's also smooth over a regular scheme, so it's regular, hence normal, hence irreducible. | |
| Oct 19, 2016 at 21:47 | comment | added | Will Chen | @AdelBETINA the family is proper here, and the base is irreducible. What's the problem? | |
| Oct 19, 2016 at 21:46 | comment | added | Adel BETINA | it is locally constant in a proper flat family (EGA III) but we need the irreducibility to say that is constant ( we need properness), we know that $X(2)$ is proper. | |
| Oct 19, 2016 at 21:44 | comment | added | Will Chen | @AdelBETINA arithmetic genus is invariant in flat families. Since our family is moreover smooth, arithmetic genus = geometric genus. | |
| Oct 19, 2016 at 21:44 | comment | added | Adel BETINA | But juste one thing, how we can proof that the genus of any fibre is zero (we know that the genus is locally constant)? it is easy to see that X(2) is irreducible ? | |
| Oct 19, 2016 at 21:43 | comment | added | Adel BETINA | if you take $X(2)$ as the normalisation over the $j$-line, it will be a Cohen-Macaulay scheme and since it is flat over $\mathbb{Z}$, it will be Cohen-Macaulay over $\mathbb{Z}$. If we can proof that the fiber at $2$ is generically reduced, it will be reduced. | |
| Oct 19, 2016 at 21:42 | comment | added | Will Chen | @AdelBETINA Though incidentally, the normalization of the projective $j$-line over $\mathbb{Z}$ inside the function field of $X(2)$ may well (for stupid reasons) just be $\mathbb{P}^1_{\mathbb{Z}}$, but if this is true, then this normalization is probably not the right model to look at from a moduli-theoretic point of view. | |
| Oct 19, 2016 at 21:37 | comment | added | Will Chen | @AdelBETINA Sorry that was a typo. In my answer I never look at the fiber over 2. In any case your question doesn't ask about the fiber over 2. | |
| Oct 19, 2016 at 21:37 | history | edited | Will Chen | CC BY-SA 3.0 |
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| Oct 19, 2016 at 21:36 | comment | added | Will Chen | @AdelBETINA I don't think that's even true, at least if you take the Katz-Mazur regular model of $M(2)$. | |
| Oct 19, 2016 at 21:36 | comment | added | Adel BETINA | Since in your answer you think that the normalisation of $X(2)$ over the j-line is $\mathbb{P}^1_{\mathbb{Z}}$ | |
| Oct 19, 2016 at 21:35 | comment | added | Adel BETINA | I want to say how to prove that the fibre is reduced at $2$? | |
| Oct 19, 2016 at 21:28 | comment | added | Will Chen | @AdelBETINA Where did I say "smooth implies irreducible"? Anyway, $X(2)$ is certainly connected. | |
| Oct 19, 2016 at 21:26 | comment | added | Adel BETINA | I forget to add in my question '' 2-nero polygone ''. In your answer you said that smooth implies irreducible, I think that is not true because we need to know that our space is connected and which is true by a result of Zariski since $(X(2)(\mathbb{C})$ is connected. Exemple $X(3)(\mathbb{C})$ is not connected but $X(3)$ is smooth. | |
| Oct 19, 2016 at 21:24 | history | edited | Will Chen | CC BY-SA 3.0 |
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| Oct 19, 2016 at 21:17 | history | answered | Will Chen | CC BY-SA 3.0 |