Rationality of flag varieties Let $X$ be a (generalised) flag variety over an algebraically closed field $k$ of characteristic zero, that is to say, $X$ is a projective variety which is a homogeneous space for some algebraic group $G$. Any such $X$ can be realised as a quotient $X=G/P$, where $P$ is a parabolic subgroup (please correct me if this is wrong!)
Some basic properties of $X$ are:


*

*$X$ is a fano variety

*$X$ is unirational


The fact that it is fano is not perhaps immediately obvious, but it is clear that it is unirational since it is dominated by $G$, which is a rational variety. There is no reason to expect your average fano variety to be rational, however are flag varieties rational? I am particularly interested in the case where $dim X = 3$.
 A: As indicated by Piotr, the answer to your question is yes.  One helpful source is the Springer graduate text by Borel, Linear Algebraic Groups; see AG.13.7 and Chapter 14, especially 14.14.  For the study of generalized flag varieties you may as well assume the group $G$ is reductive because the unipotent radical is factored out.   Here the study of the Bruhat decomposition in $G$ over any algebraically closed field leads to the fact that $G$ itself is a rational variety.    The basic idea is that $G$ has a Zariski-dense open subset which is just a product of copies of the additive and multiplicative groups of the field.  Adapting this to $G/P$ is then straightforward though not written down explicitly by Borel.   For instance, the usual flag variety $G/B$ has a dense open subset isomorphic to the product of copies of the additive group (one for each positive root); this mirrors the situation in the Lie algebra.  In $G/P$ only some of the positive roots play this role.  (As Dave Anderson indicates in his comment, the essential feature of Bruhat decomposition for flag varieties is a nice cell decomposition defined in terms of the root system and Weyl group.)
P.S. My convention here is that $B$ involves the negative roots, a convention useful for classical representation theory.    But all Borel subgroups are conjugate, so it doesn't matter.    
