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 Grassman 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 = rk(E) 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[1]
which is morally given by the second fundamental form. In particular, when 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
- Eisenbud, David; Joe, Harris (2016), 3264 and All That: A Second Course in Algebraic Geometry, C. U.P., ISBN 978-1107602724
- William 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