What is Flag bundle?

In algebraic geometry, the flag bundle of a flag

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

such that p − 1 {\displaystyle p^{-1}} is a flag V ∙ {\displaystyle V_{\bullet }} of vector spaces such that V i {\displaystyle V_{i}} is a vector subspace of x {\displaystyle _{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.