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:

:$p:\; G\_d(E)\; \backslash to\; X$

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, $G\_1(E)\; =\; \backslash mathbb\{P\}(E)$ 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

:$0\; \backslash to\; S\; \backslash to\; p^*E\; \backslash to\; Q\; \backslash to\; 0$.

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 $\backslash wedge^d\; S$ is the determinant line bundle. Now, by the universal property of a projective bundle, the injection $\backslash wedge^r\; S\; \backslash to\; p^*\; (\backslash wedge^r\; E)$ corresponds to the morphism over X:

:$G\_d(E)\; \backslash to\; \backslash mathbb\{P\}(\backslash wedge^r\; E)$,

which is nothing but a family of Plücker embeddings.

The relative tangent bundle T_Gd(E)/X of G_d(E) is given by{{harvnb|Fulton|1998|loc=Appendix B.5.8}}

:$T\_\{G\_d(E)/X\}\; =\; \backslash operatorname\{Hom\}(S,\; Q)\; =\; S^\{\backslash vee\}\; \backslash otimes\; Q,$

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 $\backslash mathbb\{P\}(V)$), there is the natural identification (see Chern class#Complex projective space for example):

:$\backslash operatorname\{Hom\}(l,\; V/l)\; =\; T\_l\; \backslash mathbb\{P\}(V)$

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:

:$0\; \backslash to\; \backslash mathcal\{O\}\_\{\backslash mathbb\{P\}(E)\}\; \backslash to\; p^*\; E\; \backslash otimes\; \backslash mathcal\{O\}\_\{\backslash mathbb\{P\}(E)\}(1)\; \backslash to\; T\_\{\backslash mathbb\{P\}(E)/X\}\; \backslash to\; 0$,

which is the relative version of the Euler sequence.