Grassmann bundle explained

In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X:

p: G _{d(E)} \to 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

0 \to S \to p ^{*E} \to Q \to 0 .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 \wedge ^{d} S is the determinant line bundle. Now, by the universal property of a projective bundle, the injection \wedge ^{r} S \to p ^{*} (\wedge ^{r} E) corresponds to the morphism over X:G _{d(E)} \to P (\wedge ^{r} E) ,which is nothing but a family of Plücker embeddings.

The relative tangent bundle T_Gd(E)/X of G_d(E) is given by

T _{G _{d(E)/X}} = \operatorname{Hom}(S, Q) = S ^{\vee} ⊗ Q, 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 l in V passing through the origin (a point of P (V) ), there is the natural identification (see Chern class#Complex projective space for example):\operatorname{Hom}(l, V/l) = T _{l} P (V) and the above is the family-version of this identification. (The general care is a generalization of this.)

In the case d = 1, the early exact sequence tensored with the dual of S = O(-1) gives:

0 \to l{O} _{P(E)} \to p ^{*} E ⊗ l{O} _{P(E)} ( 1 ) \to T _{P(E)/X} \to 0 ,which is the relative version of the Euler sequence