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In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X:
such that the fiber p − 1 = G d {\displaystyle p^{-1}=G_{d}} is the Grassmannian of the d-dimensional vector subspaces of E x {\displaystyle E_{x}}. For example, G 1 = P {\displaystyle G_{1}=\mathbb {P} } is the projective bundle of E. In the other direction, a Grassmann bundle is a special case of a 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
Specifically, if V is in the fiber p, then the fiber of S over V is V itself; thus, S has rank r = rk and ∧ r S {\displaystyle \wedge ^{r}S} is the determinant line bundle. Now, by the universal property of a projective bundle, the injection ∧ r S → p ∗ {\displaystyle \wedge ^{r}S\to p^{*}} corresponds to the morphism over X: