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Jul 1, 2018 at 15:35 comment added cccp Dear Jason. I think $(1)$ is clear no ? I mean $V_i$ is the irreducible representation associated to a non trivial dominant weight. In any case thank you very much for your answer about $(2)$ (it will take some time for me to parse since I don't know some of the notions you are using).
Jul 1, 2018 at 15:17 comment added Jason Starr You are ask several questions: (1) why is $V_i$ nonzero, (2) why is the morphism $f:G/B \to\prod_i \mathbb{P}(V_i)$ an embedding, (3) why is the image of $f$ defined by the relations above? Let me address only (2), assuming (1). First, the cone of effective curve classes on $G/B$ is spanned by the "Demazure curves", i.e., the fibers of the $\mathbb{P}^1$-bundles $G/B\to G/P_i$ used by Demazure in his "simple proof" of Borel-Weil-Bott. The fundamental weights span the dual cone, so the fibers of $f$ contain no curves, i.e., they are finite. Now use homogeneity and simple connectedness.
Jul 1, 2018 at 15:11 history edited cccp CC BY-SA 4.0
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Jul 1, 2018 at 14:07 history asked cccp CC BY-SA 4.0