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:

such that the fiber ${\displaystyle p^{-1}(x)=G_{d}(E_{x})}$ is the Grassmannian of the d-dimensional vector subspaces of ${\displaystyle E_{x}}$. For example, ${\displaystyle G_{1}(E)=\mathbb {P} (E)}$ 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.

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