Flag bundle
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In algebraic geometry, the flag bundle of a flag[1]
- [math]\displaystyle{ E_{\bullet}: E = E_l \supsetneq \cdots \supsetneq E_1 \supsetneq 0 }[/math]
of vector bundles on an algebraic scheme X is the algebraic scheme over X:
- [math]\displaystyle{ p: \operatorname{Fl}(E_{\bullet}) \to X }[/math]
such that [math]\displaystyle{ p^{-1}(x) }[/math] is a flag [math]\displaystyle{ V_{\bullet} }[/math] of vector spaces such that [math]\displaystyle{ V_i }[/math] is a vector subspace of [math]\displaystyle{ (E_i)_x }[/math] 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
- ↑ Here, [math]\displaystyle{ E_i }[/math] is a subbundle not subsheaf of [math]\displaystyle{ E_{i+1}. }[/math]
- William Fulton. (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4
- Expo. VI, § 4. of Berthelot, Pierre, ed (1971) (in fr). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225). Berlin; New York: Springer-Verlag. pp. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8.
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