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 $dim X = 3$.