Flag bundle

In algebraic geometry, the flag bundle of a flag
 * $$E_{\bullet}: E = E_l \supsetneq \cdots \supsetneq E_1 \supsetneq 0$$

of vector bundles on an algebraic scheme X is the algebraic scheme over X:
 * $$p: \operatorname{Fl}(E_{\bullet}) \to X$$

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.