# Grassmann bundle

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In algebraic geometry, the **Grassmann *d*-plane bundle** of a vector bundle *E* on an [algebraic scheme](/source/Algebraic_scheme) *X* is a scheme over *X*:

- p : G d ( E ) → X {\displaystyle p:G_{d}(E)\to X}

such that the fiber p − 1 ( x ) = G d ( E x ) {\displaystyle p^{-1}(x)=G_{d}(E_{x})} is the [Grassmannian](/source/Grassmannian) of the *d*-dimensional vector subspaces of E x {\displaystyle E_{x}} . For example, G 1 ( E ) = P ( E ) {\displaystyle G_{1}(E)=\mathbb {P} (E)} is the [projective bundle](/source/Projective_bundle) of *E*. In the other direction, a Grassmann bundle is a special case of a (partial) [flag bundle](/source/Flag_bundle). Concretely, the Grassmann bundle can be constructed as a [Quot scheme](/source/Quot_scheme).

The Grassmann bundle of the [tangent bundle](/source/Tangent_bundle) is the [contact bundle](/source/Contact_bundle).

Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or [tautological subbundle](/source/Tautological_subbundle) *S* and universal quotient bundle *Q* that fit into

- 0 → S → p ∗ E → Q → 0 {\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 ∧ d S {\displaystyle \wedge ^{d}S} is the [determinant line bundle](/source/Determinant_line_bundle). Now, by the universal property of a projective bundle, the injection ∧ r S → p ∗ ( ∧ r E ) {\displaystyle \wedge ^{r}S\to p^{*}(\wedge ^{r}E)} corresponds to the morphism over *X*:

- G d ( E ) → P ( ∧ r E ) {\displaystyle G_{d}(E)\to \mathbb {P} (\wedge ^{r}E)} ,

which is nothing but a family of [Plücker embeddings](/source/Pl%C3%BCcker_embedding).

The [relative tangent bundle](/source/Relative_tangent_bundle) *T**G**d*(*E*)/*X* of *G**d*(*E*) is given by[1]

- T G d ( E ) / X = Hom ⁡ ( S , Q ) = S ∨ ⊗ Q , {\displaystyle T_{G_{d}(E)/X}=\operatorname {Hom} (S,Q)=S^{\vee }\otimes Q,}

which morally is given by the [second fundamental form](/source/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 {\displaystyle l} in *V* passing through the origin (a point of P ( V ) {\displaystyle \mathbb {P} (V)} ), there is the natural identification (see [Chern class#Complex projective space](/source/Chern_class#Complex_projective_space) for example):

- Hom ⁡ ( l , V / l ) = T l P ( V ) {\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:

- 0 → O P ( E ) → p ∗ E ⊗ O P ( E ) ( 1 ) → T P ( E ) / X → 0 {\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](/source/Euler_sequence).

## References

1. **[^](#cite_ref-1)** [Fulton 1998](#CITEREFFulton1998), Appendix B.5.8

- Eisenbud, David; Joe, Harris (2016), *3264 and All That: A Second Course in Algebraic Geometry*, C. U.P., [ISBN](/source/ISBN_(identifier)) [978-1107602724](https://en.wikipedia.org/wiki/Special:BookSources/978-1107602724)

- Fulton, William (1998), *Intersection theory*, [Ergebnisse der Mathematik und ihrer Grenzgebiete](/source/Ergebnisse_der_Mathematik_und_ihrer_Grenzgebiete). 3. Folge., vol. 2 (2nd ed.), Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [ISBN](/source/ISBN_(identifier)) [978-3-540-62046-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-62046-4), [MR](/source/MR_(identifier)) [1644323](https://mathscinet.ams.org/mathscinet-getitem?mr=1644323)

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