Flag bundle

In algebraic geometry, the flag bundle of a flag^[1]

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

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.

Construction

A flag bundle can be constructed inductively.

References

1. Here, ${\displaystyle E_{i}}$ is a subbundle not subsheaf of ${\displaystyle E_{i+1}.}$

• 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). Vol. 225. Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.