In algebraic geometry, the flag bundle of a flagHere,

E_i

is a subbundle not subsheaf of

E_.

E_: 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(E_) \to X

such that

p^(x)

is a flag

V_

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 In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smo ...

and if the length of the flag is one, then it is the

Grassmann bundle In algebraic geometry, the Grassmann ''d''-plane bundle of a vector bundle ''E'' on an algebraic scheme ''X'' is a scheme over ''X'': :p: G_d(E) \to X such that the fiber p^(x) = G_d(E_x) is the Grassmannian of the ''d''-dimensional vector subspace ...

; 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 Algebraic geometry {{algebraic-geometry-stub