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

which is nothing but a family of Plücker embeddings.

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

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

:\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:

: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

{{Reflist}}

  • {{citation|first=David|last=Eisenbud|first2=Harris|last2=Joe|title=3264 and All That: A Second Course in Algebraic Geometry|publisher=C. U.P.|year=2016|isbn=978-1107602724}}
  • {{Citation | title=Intersection theory | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-62046-4 | mr=1644323 | year=1998 | volume=2 | edition=2nd | first=William |last=Fulton }}

Category:Algebraic geometry