Grassmann bundle
In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X:
:
such that the fiber is the Grassmannian of the d-dimensional vector subspaces of . For example, is the projective bundle of E. In the other direction, a Grassmann bundle is a special case of a (partial) flag bundle. Concretely, the Grassmann bundle can be constructed as a Quot scheme.
Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or tautological subbundle S and universal quotient bundle Q that fit into
:.
Specifically, if V is in the fiber p−1(x), then the fiber of S over V is V itself; thus, S has rank r = d = dim(V) and is the determinant line bundle. Now, by the universal property of a projective bundle, the injection corresponds to the morphism over X:
:,
which is nothing but a family of Plücker embeddings.
The relative tangent bundle TGd(E)/X of Gd(E) is given by{{harvnb|Fulton|1998|loc=Appendix B.5.8}}
:
which morally is given by the second fundamental form. In the case d = 1, it is given as follows: if V is a finite-dimensional vector space, then for each line in V passing through the origin (a point of ), there is the natural identification (see Chern class#Complex projective space for example):
:
and the above is the family-version of this identification. (The general care is a generalization of this.)
In the case d = 1, the early exact sequence tensored with the dual of S = O(-1) gives:
:,
which is the relative version of the Euler sequence.
References
{{Reflist}}
- {{citation|first=David|last=Eisenbud|first2=Harris|last2=Joe|title=3264 and All That: A Second Course in Algebraic Geometry|publisher=C. U.P.|year=2016|isbn=978-1107602724}}
- {{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 | first=William |last=Fulton }}