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
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 = rk(E) and
which is nothing but a family of Plücker embeddings.
The relative tangent bundle TGd(E)/X of Gd(E) is given by
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.