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



Flag Varieties - Projective Curves
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](https://mathoverflow.net/questions/3296/projective-curves-which-are-principal-bundles/3301#3301).)


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 $1$.


