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:

${\displaystyle p:G_{d}(E)\to X}$

such that the fiber ${\displaystyle p^{-1}(x)=G_{d}(E_{x})}$ is the Grassmannian of the d-dimensional vector subspaces of ${\displaystyle E_{x}}$. For example, ${\displaystyle G_{1}(E)=\mathbb {P} (E)}$ 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 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

${\displaystyle 0\to S\to p^{*}E\to Q\to 0}$.

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

${\displaystyle G_{d}(E)\to \mathbb {P} (\wedge ^{r}E)}$,

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]

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

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 ${\displaystyle l}$ in V passing through the origin (a point of ${\displaystyle \mathbb {P} (V)}$), there is the natural identification (see Chern class#Complex projective space for example):

${\displaystyle \operatorname {Hom} (l,V/l)=T_{l}\mathbb {P} (V)}$

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:

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

which is the relative version of the Euler sequence.

References

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