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:

[math]\displaystyle{ p: G_d(E) \to X }[/math]

such that the fiber [math]\displaystyle{ p^{-1}(x) = G_d(E_x) }[/math] is the Grassmannian of the d-dimensional vector subspaces of [math]\displaystyle{ E_x }[/math]. For example, [math]\displaystyle{ G_1(E) = \mathbb{P}(E) }[/math] 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

[math]\displaystyle{ 0 \to S \to p^*E \to Q \to 0 }[/math].

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

[math]\displaystyle{ G_d(E) \to \mathbb{P}(\wedge^r E) }[/math],

which is nothing but a family of Plücker embeddings.

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

[math]\displaystyle{ T_{G_d(E)/X} = \operatorname{Hom}(S, Q) = S^{\vee} \otimes Q, }[/math]

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 [math]\displaystyle{ l }[/math] in V passing through the origin (a point of [math]\displaystyle{ \mathbb{P}(V) }[/math]), there is the natural identification (see Chern class for example):

[math]\displaystyle{ \operatorname{Hom}(l, V/l) = T_l \mathbb{P}(V) }[/math]

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:

[math]\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 }[/math],

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., 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4