Homogeneous variety
From Wikipedia, the free encyclopedia
In algebraic geometry, a homogeneous variety is an algebraic variety on which an algebraic group acts transitively.[1][2] Homogeneous varieties over an algebraically closed field are quotient varieties G/H where G is an algebraic group and H a subgroup scheme (for instance, an algebraic subgroup).[3]
Such varieties are always smooth quasi-projective varieties.
Classical examples are flag varieties (when G is semisimple and H a parabolic subgroup), or more generally homogeneous spherical varieties. Severi-Brauer varieties are examples of homogeneous varieties over a field without any rational points.
