Let $X$ be an homogeneous projective variety, written as the quotient $G/P$, where $G$ is a Lie group and $P$ is a parabolic subgroup of it. It seems it is well-known that the monoid of effective curves on $X$, as a submonoid of $H_2(X, \mathbb{Z})$, is generated by finitely many curves $\beta_1, \ldots, \beta_p$. Moreover the $\beta_i$ are embedded $\mathbb{P}^1$'s in $X$. For instance in the expository notes http://arxiv.org/abs/alg-geom/9608011, the authors use this fact (e.g. at the beginning of p. 37). The question is simply: how does one prove this fact? Is there a reference for a precise proof of it (perhaps in a slightly different generality, or proving the same statement for the effective monoid inside the Chow group, instead of homology)?
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$\begingroup$ This probably isn't the right approach or the simplest, but are these $G/P$ log Fano, can that be used? $\endgroup$– Karl SchwedeCommented Sep 29, 2010 at 16:00
1 Answer
One place this is treated is in Brion's notes, http://arxiv.org/abs/math/0410240, particularly Section 1.4, and the references in the notes at the end of that section.
A little more precisely, one shows that a divisor (line bundle) is nef iff it is globally generated, and this cone is generated by the Schubert divisors. Using the fact that the Schubert classes form a self-dual basis for $H^*(X,{\Bbb Z})$ (as well as $A^*X$) under the intersection pairing, it follows that the "dual classes" to the Schubert divisors generate the cone of curves. (As with many such things in $G/P$ world, the arguments for cohomology and Chow groups are exactly the same.)
By the way, it's an easy consequence of the existence of a dense open $B$-orbit that the pseudo-effective cone is generated by the complement of the open orbit --- that is, again by the Schubert divisors. So for $G/P$, the nef and pseudo-effective cones are the same.
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$\begingroup$ Thank you very much for your answer. Your answer is about flag varieties, and I am very satisfied with that, but the question was about homogeneous variety. Is there a way to fix this discrepancy? $\endgroup$– CalcCommented Sep 30, 2010 at 6:54
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$\begingroup$ Ah, although the notes I referenced set things up for $GL_n/B$, they deliberately use language and notation so that essentially every statement applies equally well to any $G/P$. So there's actually no discrepancy. I don't know an appropriate reference that does the general $G/P$ case in an introductory way, but you could try Kumar's book "Kac-Moody groups,...". (Also, as a matter of terminology, it's common to use the terms "flag variety", "homogeneous variety", and "$G/P$" interchangeably, at least when talking about projective homogeneous spaces.) $\endgroup$ Commented Sep 30, 2010 at 15:03