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.

References

• William Fulton. (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
• Expo. VI, § 4. of Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. (1971). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225) (in French). Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.