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In algebraic geometry, the flag bundle of a flagcite-ref-1[1]

E ∙ : E = E l ⊋ ⋯ ⊋ E 1 ⊋ 0 {\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:

p : Fl ⁡ ( E ∙ ) → X {\displaystyle p:\operatorname {Fl} (E_{\bullet })\to X}

such that p − 1 ( x ) {\displaystyle p^{-1}(x)} is a flag V ∙ {\displaystyle V_{\bullet }} of vector spaces such that V i {\displaystyle V_{i}} is a vector subspace of ( E i ) x {\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

• References

A flag bundle can be constructed inductively.

cite-note-11. ↑ Here, E i {\displaystyle E_{i}} is a subbundle not subsheaf of E i + 1 . {\displaystyle E_{i+1}.}

• citerefwilliam-fulton-1998William 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 citerefberthelotalexandre-grothendieckluc-illusie1971Berthelot, 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.