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 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, then the fiber of S over V is V itself; thus, S has rank r = rk 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/X of Gd is given by
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, there is the natural identification :
and the above is the family-version of this identification.
In the case d = 1, the early exact sequence tensored with the dual of S = O gives:
which is the relative version of the Euler sequence.
