Timeline for (non-trivial) isotrivial family of elliptic curves over C^{\times}
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 2, 2011 at 12:41 | comment | added | Qfwfq | Maybe Hugo was looking for a completely elementary and self contained proof. | |
| Sep 2, 2011 at 12:37 | comment | added | rita | The last statement is a well known fact, have a look at Beauville's book on surfaces, the chapter on birational maps of surfaces. | |
| Sep 2, 2011 at 12:12 | vote | accept | Hugo Chapdelaine | ||
| Sep 2, 2011 at 14:10 | |||||
| Sep 2, 2011 at 12:11 | comment | added | Hugo Chapdelaine | Of course, one could verify by a local computation that a blow up at one point does not change $h^1(\mathcal{O})$ but then one would have to show that a birational map between $X_1$ and $X_2$ could be obtained by a sequence of blow ups which a priori looks as a more difficult question than the original. | |
| Sep 2, 2011 at 12:10 | comment | added | Hugo Chapdelaine | I'm quite happy with your proof but I think there should be a more elementary proof which is self contained. You see the whole point of this question is to come up with a birational invariant and if we assume from the outset that $h^1(\mathcal{O})$ is a birational invariant then it (almost) kills the problem. | |
| Sep 2, 2011 at 8:36 | comment | added | rita | @Hugo (continued): there is also a famous rationality criterion by Castelnuovo, that says that a surface $S$ is rational iff $h^1({\mathcal O})=h^0(2K_S)=0$. | |
| Sep 2, 2011 at 6:27 | comment | added | rita | @Georges: I've edited and fixed the typos (I hope). Thanks! @Hugo: $h^1({\mathcal O})$ is a birational invariant because by Hodge theory it is equal to $h^0(\Omega^1) | |
| Sep 2, 2011 at 6:21 | history | edited | rita | CC BY-SA 3.0 |
edited body
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| Sep 2, 2011 at 1:18 | comment | added | Hugo Chapdelaine | So I guess the first $X_2$ should read as $X_1$. Is it completely obvious that $h^1(\mathcal{O}_X)$ is a birational invariant? After all $\mathcal{O}_X$ is the sheaf of regular functions on $X$ which a priori could change under birational maps. | |
| Sep 1, 2011 at 21:30 | history | answered | rita | CC BY-SA 3.0 |