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Is it true that there are no projective curves which are also flag manifolds? If so, why?

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up vote 10 down vote accepted

The projective line is both a curve of genus $0$ and a flag variety for $SL_2$. This is the only example. This is true for about a zillion reasons:

Flag varieties are rational (because of the Bruhat decomposition.) Curves, of genus $>0$, are not.

Flag varieties have transitive group actions. Curves of genus $\geq 2$ do not (see my answer here.)

If you look at the classification of flag varieties, there are only finitely many of any given dimension. In paritcular, there is only one example of dimension $1$.

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I'm having a little trouble phrasing this politely: What definition of flag variety do you know, and what definition of projective line, so that it is not clear that the projective line is an example of a flag variety? – David Speyer Nov 9 '09 at 20:20

According to this:

"Flag varieties are naturally projective varieties."

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Yes, I've seen this. But my question is whether they are projective CURVES. – Jean Delinez Nov 9 '09 at 19:32

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