Timeline for Quotients of rational surfaces
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 12, 2013 at 17:33 | vote | accept | Jérémy Blanc | ||
| Jan 11, 2013 at 16:53 | history | edited | Jason Starr | CC BY-SA 3.0 |
Added reference to "Cubic Forms"
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| Jan 11, 2013 at 16:33 | comment | added | Jason Starr | @Jérémy: I double-checked in "Cubic Forms". Actually a del Pezzo of degree $4$ with a rational point is not automatically rational, and this gives examples as you ask, cf. Theorem IV.29.2, Theorm IV.29.4 and Remark IV.29.4.1, pp. 157-158 with $r=5$, and also Section IV.31, pp. 174--182. | |
| Jan 11, 2013 at 14:52 | comment | added | Jason Starr | Certainly you can find examples as above where the field is $\mathbb{Q}(t)$, but I guess you want the field to be $\mathbb{Q}$. I believe every del Pezzo surface of degree at least $4$ with a rational point is rational. So that suggests to focus on cubic surfaces. In Manin's "Cubic Forms", I believe he states a condition regarding the Galois action on the Picard lattice that implies that the cubic surface is irrational. So that is a place to start. | |
| Jan 11, 2013 at 13:05 | comment | added | Jérémy Blanc | Thanks for the nice answer. Do you think that such a phenomenon could also be found over $\mathbb{Q}$? | |
| Jan 10, 2013 at 18:27 | history | edited | Jason Starr | CC BY-SA 3.0 |
Clarified connection with Clemens-Griffiths.
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| Jan 10, 2013 at 16:00 | history | answered | Jason Starr | CC BY-SA 3.0 |