Flag variety

Remarks

The group $\mathrm{GL}(V)$ acts transitively on the flags of a given type, so the flag variety is a homogeneous space. The stabilizer of a full flag is a Borel subgroup, conjugate to the upper triangular matrices, so the full flag variety is $\mathrm{GL}_n / B$.

General form: let $G$ be a connected reductive group over an algebraically closed field. A closed subgroup $P$ of $G$ is parabolic if it contains a Borel subgroup. The generalized flag variety is $G / P$, a smooth projective homogeneous variety.

Example: for $m = 1$, the flag variety $\mathrm{Fl}(d_1; V)$ is the Grassmannian $\mathrm{Gr}(d_1, V)$, and for $d_1 = 1$ it is the projective space $\mathbb{P}(V)$.

Non-example: let $T$ be the diagonal torus of $\mathrm{GL}_2$, the stabilizer of the pair of coordinate lines. The quotient $\mathrm{GL}_2 / T$ is the space of ordered pairs of distinct lines in $k^2$, which is $\mathbb{P}^1 \times \mathbb{P}^1$ minus its diagonal. It is not projective, since $T$ contains no Borel subgroup, and it is not a flag variety.