Flag bundle explained
In algebraic geometry, the flag bundle of a flag[1]
E\bullet:E=El\supsetneq … \supsetneqE1\supsetneq0
of vector bundles on an algebraic scheme
X is the algebraic scheme over
X:
p:\operatorname{Fl}(E\bullet)\toX
such that
is a flag
of vector spaces such that
is a vector subspace of
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
- Expo. VI, § 4. of Book: Berthelot . Pierre . Pierre Berthelot (mathematician) . . . Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225) . 1971 . . Berlin; New York . fr . xii+700 . true . 10.1007/BFb0066283 . 978-3-540-05647-8 . 0354655.
Notes and References
- Here,
is a subbundle not subsheaf of