
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 formula_2 is the Grassmannian of the "d"-dimensional vector subspaces of formula_3. For example, formula_4 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
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" = rk("E") and formula_6 is the determinant line bundle. Now, by the universal property of a projective bundle, the injection formula_7 corresponds to the morphism over "X":
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
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 formula_10 in "V" passing through the origin (a point of formula_11), there is the natural identification (see Chern class#Complex projective space for example):
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:
which is the relative version of the Euler sequence.