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 $p^(x) = G_d(E_x)$ is the Grassmannian of the ''d''-dimensional vector subspaces of $E_x$. For example, $G_1(E) = \mathbb(E)$ 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'':
:$G_d(E) \to \mathbb(\wedge^r E)$,
which is nothing but a family of Plücker embeddings.

The relative tangent bundle ''T''_{''G''''d''(''E'')/''X''} of ''G''_''d''(''E'') is given by
:$T_ = \operatorname(S, Q) = S^ \otimes Q,$
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 $\mathbb(V)$), there is the natural identification (see Chern class#Complex projective space for example):
:$\operatorname(l, V/l) = T_l \mathbb(V)$
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:
:$0 \to \mathcal_ \to p^* E \otimes \mathcal_(1) \to T_ \to 0$,
which is the relative version of the Euler sequence.

References

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* {{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

Algebraic geometry