Timeline for Questions Suggested by the Parabolic Subgroup Definition
Current License: CC BY-SA 2.5
5 events
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| Jun 1, 2010 at 15:35 | comment | added | BCnrd | Victor, nice observation! The coset space $G/P$ always descends to a smooth proper scheme over the entirety of $\mathbf{Z}$, no hypotheses needed on the center. But proving that a scheme over $\mathbf{C}$ doesn't admit a descent to a smooth proper $\mathbf{Z}$-scheme seems to require more than verifying that a particular $\mathbf{Q}$-descent doesn't have everywhere good reduction. So is the rigorous proof of your surface example nonetheless easy? | |
| May 31, 2010 at 4:22 | comment | added | Victor Protsak | Two important properties that $G/P$ homogeneous spaces have are (1) they are defined over $\mathbb{Z}$ and (2) they have good reduction everywhere (possibly, with modulo assumptions on the center of $G$). Already for del Pezzo surfaces obtained by blowing up a few points on $\mathbb{P}^2$, (2) fails. | |
| May 31, 2010 at 0:20 | comment | added | S. Carnahan♦ | That sounds like a correct reference, although I have never read the paper itself. I'm inclined to agree that the class of Fano varieties is itself far larger than the class of varieties of the form G/P, but I don't know any more adjectives (except "homogeneous" which is cheating) that narrow things down correctly. | |
| May 30, 2010 at 22:07 | comment | added | Jim Humphreys |
I guess this theorem comes from the two-part paper by Borel and Hirzebruch on homogeneous spaces and characteristic classes published in 1958-59? In any case, smooth projective varieties (or compact complex manifolds) of the form $G/P$ are scarce thanks to the Borel-Chevalley structure theory and the associated root system data. Even for smooth Schubert varieties, I'd expect few other than flag varieties to be of this form but can't quantify this.
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| May 30, 2010 at 18:05 | history | answered | S. Carnahan♦ | CC BY-SA 2.5 |