Flag bundle

In algebraic geometry, the flag bundle of a flag

$E_{\bullet }:E=E_l⊋\cdots ⊋E_1⊋0$

of vector bundles on an algebraic scheme X is the algebraic scheme over X:

$p:\mathrm{Fl}(E_{\bullet })\to X$

such that $p^{-1}(x)$ is a flag $V_{\bullet }$ of vector spaces such that $V_i$ is a vector subspace of $(E_i{)}_x$ of dimension i.

If X is a point, then a flag bundle is a flag variety and if the length of the flag is one, then it is the Grassmann bundle; hence, a flag bundle is a common generalization of these two notions.

Construction

A flag bundle can be constructed inductively.

References

• Expo. VI, § 4. of