flag bundle

In algebraic geometry, the flag bundle of a flagHere, $E_i$ is a subbundle not subsheaf of $E_{i+1}.$

: $E_{\bullet}: E = E_l \supsetneq \cdots \supsetneq E_1 \supsetneq 0$

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

{{Citation | title=Intersection theory | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-62046-4 | mr=1644323 | year=1998 | volume=2 | edition=2nd | author=William Fulton.}}
Expo. VI, § 4. of {{cite book

| editor-last = Berthelot

| editor-first = Pierre

| editor-link = Pierre Berthelot (mathematician)

| editor2=Alexandre Grothendieck

| editor3=Luc Illusie

| title = Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225)

| year = 1971

| volume = 225

| publisher = Springer-Verlag

| location = Berlin; New York

| language = fr

| pages = xii+700

| no-pp = true

|doi=10.1007/BFb0066283

|isbn= 978-3-540-05647-8

| mr = 0354655

}}

Category:Algebraic geometry