Timeline for Classification of smooth algebraic surfaces with a smooth morphism to $\Bbb P^1$
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 16, 2018 at 22:15 | history | edited | Ariyan Javanpeykar | CC BY-SA 4.0 |
deleted 3 characters in body
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| Oct 16, 2018 at 20:00 | vote | accept | Zhiyu | ||
| Oct 16, 2018 at 17:53 | history | edited | Ariyan Javanpeykar | CC BY-SA 4.0 |
Made things a bit more clear hopefully.
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| Oct 16, 2018 at 8:35 | comment | added | Daniel Loughran | For $g=0$ there is a section by Tsen's theorem. So you get a $\mathbb{P}^1$-bundle over $\mathbb{P}^1$ with a section, which is thus isomorphic to a Hirzebruch surfaces (so not necessarily trivial). | |
| Oct 16, 2018 at 3:22 | comment | added | Zhiyu | Thanks, the tool of moduli space is wonderful. Last but not least, what about the case $g=0$? | |
| Oct 15, 2018 at 23:27 | comment | added | Will Sawin | If $g=1$ then the automorphism of the Jacobean, as a pointed elliptic curve, are finite so the same argument shows isotrivial implies trivial. So we have to classify $H^1(\mathbb P^1, E)$ for elliptic $E$. Use that it is algebraic to find a multi section, hence show it’s torsion, then use the vanishing of $ H^1(\mathbb P^, E[n])$ to conclude. Without algebraicity, the Hopf surface is a counterexample. | |
| Oct 15, 2018 at 18:24 | history | edited | Ariyan Javanpeykar | CC BY-SA 4.0 |
added 328 characters in body
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| Oct 15, 2018 at 18:12 | history | answered | Ariyan Javanpeykar | CC BY-SA 4.0 |