Timeline for Moduli of flag varieties
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 27, 2014 at 8:15 | comment | added | Ariyan Javanpeykar | @pmath. Flag varieties are examples of Fano varieties. Thus they live in the moduli of Fano varieties, i.e., "anti-canonically polarized varieties". Although the moduli of flag varieties is zero-dimensional, the moduli of Fano's is positive dimensional in general. Consider, for example, Del Pezzo surfaces of degree 1,2,3, or 4. | |
| Feb 25, 2014 at 17:56 | comment | added | eventually | Is there a "bigger moduli space" where the points associated to the moduli of flags live "naturally" ? | |
| Feb 24, 2014 at 18:16 | vote | accept | Daniel Loughran | ||
| Feb 24, 2014 at 4:40 | comment | added | Jason Starr | ... which equals $6(p^{2n}-9)$. | |
| Feb 24, 2014 at 4:37 | comment | added | Jason Starr | Last comment: "genus" --> "$(K_X)_X^3$". | |
| Feb 24, 2014 at 4:17 | comment | added | Jason Starr | @DaveAnderson: "aren't all those isomorphic to $SL_3/B$?" No, you can compute the genus using the genus formula. They all have different genera, hence they are all mutually non-isomorphic. However, they are related by a sequence of purely inseparable morphisms. | |
| Feb 24, 2014 at 0:40 | comment | added | Dave Anderson | @Jason, neat example --- aren't those all isomorphic to $SL_3/B$? | |
| Feb 24, 2014 at 0:14 | comment | added | Dave Anderson | @Daniel, Ben Webster has already given a slick conceptual proof, but at least in this setting (alg closed, char 0) an easy direct proof comes by treating semisimple $G$'s as a list of examples. You can just use Will's/Jim's suggestion. (After reducing the problem from semisimple to simple there are only three (or four) infinite series of flag varieties: $SL_n/P$, $Sp_{2n}/P$, and $SO_n/P$. By direct inspection, each of these has dimension $\to \infty$ as $n\to\infty$.) | |
| Feb 23, 2014 at 23:39 | comment | added | Jason Starr | @JimHumphreys: Yeah, I know all that. | |
| Feb 23, 2014 at 23:07 | comment | added | Jim Humphreys | @Jason: In the question, "homogeneous space" has the usual meaning for linear algebraic groups: a quasi-projective variety isomorphic to $G/H$, where $H$ is the (closed) isotropy group of a point and all such subgroups are conjugate in $G$. For $G$ semisimple (or reductive), $G/H$ is projective iff $H$ is parabolic, and is then called a relative or generalized flag variety. This doesn't depend on the characteristic. In particular, homogeneous spaces have dimensions bounded by the dimension of $G$. | |
| Feb 23, 2014 at 20:11 | comment | added | Jason Starr | Ridiculous observation: in positive characteristic $p$, for every integer $n$, the following smooth, projective, $3$-dimensional variety, $\{([X_0,X_1,X_2],[Y_0,Y_1,Y_2])\in \mathbb{P}^2\times \mathbb{P}^2 : X_0Y_0^{p^n} + X_1Y_1^{p^n} + X_2Y_2^{p^n}\}$, is homogeneous for an action of $\textbf{SL}_3$. | |
| Feb 23, 2014 at 18:57 | comment | added | Jim Humphreys | @Daniel: Though you may be interested just in charcteristic 0, everything here generalizes immediately to arbitrary characteristic since the dimensions of (generalized) flag varieties depend only on the root system or Dynkin diagram. That's seen in all textbook treatments of semisimple algebraic groups and their parabolic subgroups. | |
| Feb 23, 2014 at 18:49 | answer | added | Ben Webster♦ | timeline score: 9 | |
| Feb 23, 2014 at 18:39 | comment | added | Daniel Loughran | The thing that is worrying me is something like the possible existence of an infinite sequence $P_i \subset G_i$ ($i=1,\ldots,\infty$) where $\dim G_i \to \infty$, but $\dim G_i/P_i = n$ is fixed. In which case one needs to show that there are only finitely many isomorphism classes of varieties in the set $\{G_i/P_i: i=1,\ldots,\infty\}$. Does your approach achieve this? | |
| Feb 23, 2014 at 18:15 | comment | added | Will Sawin | Yes. Use classification of semisimple algebraic groups in terms of Dynkin diagrams + classification of parabolic subgroups in terms of subsets of Dynkin diagrams. Lower bound dimension of flag variety in terms of dimension of group, giving finiteness. | |
| Feb 23, 2014 at 18:13 | history | asked | Daniel Loughran | CC BY-SA 3.0 |