Flag bundle

In algebraic geometry, the flag bundle of a flag

${\displaystyle E_{\bullet }:E=E_{l}\supsetneq \cdots \supsetneq E_{1}\supsetneq 0}$

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

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

such that ${\displaystyle p^{-1}(x)}$ is a flag ${\displaystyle V_{\bullet }}$ of vector spaces such that ${\displaystyle V_{i}}$ is a vector subspace of ${\displaystyle (E_{i})_{x}}$ 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.