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 p − 1 ( x ) = G d ( E x ) is the Grassmannian of the d-dimensional vector subspaces of E x . For example, G 1 ( E ) = P ( E ) 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 = rk(E) and ∧ r S is the determinant line bundle. Now, by the universal property of a projective bundle, the injection ∧ r S → p ∗ ( ∧ r E ) corresponds to the morphism over X:

which is nothing but a family of Plücker embeddings.

The relative tangent bundle T_Gd(E)/X of G_d(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.