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:

$p: G_d(E) \to X$

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, $G_1(E) = \mathbb{P}(E)$ 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 Grassman 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 = rk(E) and $\wedge^r S$ 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:

$G_d(E) \to \mathbb{P}(\wedge^r E)$,

which is nothing but a family of Plücker embeddings.

The relative tangent bundle T_Gd(E)/X of G_d(E) is given by^[1]

$T_{G_d(E)/X} = \operatorname{Hom}(S, Q) = S^{\vee} \otimes Q,$

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:

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

which is the relative version of the Euler sequence.

References

1. Fulton Appendix B.5.8

• Eisenbud, David; Joe, Harris (2016), 3264 and All That: A Second Course in Algebraic Geometry, C. U.P., ISBN 978-1107602724
• William Fulton. (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323