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)?



Mori cone of homogeneous varieties
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.


