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
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
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 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
), there is the natural identification (see Chern class#Complex projective space for example): 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(-1) gives: which is the relative version of the Euler sequence.