By Carlos Contou-Carrere

The first a part of this booklet introduces the Schubert Cells and kinds of the overall linear staff Gl (k^(r+1)) over a box okay in line with Ehresmann geometric approach. soft resolutions for those kinds are developed by way of Flag Configurations in k^(r+1) given by way of linear graphs referred to as minimum Galleries. within the moment half, Schubert Schemes, the common Schubert Scheme and their Canonical soft solution, when it comes to the prevalence relation in a titties relative development are built for a Reductive workforce Scheme as in Grothendieck's SGAIII. it is a subject the place algebra and algebraic geometry, combinatorics, and crew concept have interaction in strange and deep ways.

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**Example text**

Grassm (η)x , Drapm (η)x ) identifies with V (ηx ). Grassm (ηx ), Drapm (ηx )). Moreover: “if X is a k-smooth variety (resp. a k-integral variety), it follows that V (η) (resp. k-integral variety)”. 9. 9 define the embedding of Drapm (k E ) in a product of grassmannians in terms of the corresponding Plücker coordinates. Denote by pβ : Drapn (k E ) −→ Grassnβ (k E ) the canonical projection, and by p∗β (ξnβ ) the pull-back of the tautological module ξnβ . Define the E by: tautological flag ξn of submodules of ODrap E n (k ) E ξn = (p∗1 (ξn1 ) · · · ⊂ p∗l (ξnl ) ⊂ ODrap E ).

One has defined the canonical coordinates in UD (cf. 20). In fact, this parametrization corresponds to the parametrization of UD in terms of the principal action of U (Dopp ) on UD (cf. 37). One may embed M(Hβ+1 −Hβ )×Hβ in ME×E by completing a (Hβ+1 −Hβ )× Hβ -matrix m with zeros in a E × E-matrix m , which one also denotes by m. Write u(m) = 1E +m ∈ U (((Hβ+1 −Hβ ) ⊂ Hβ+1 )). There is an isomorphism: M(Hβ+1 −Hβ )×Hβ (k) −→ uD : A D = 1 β l U (((Hβ+1 − Hβ ) ⊂ Hβ )) 1 β l U (Dopp ) , defined by uD : (mβ ) → uD ((mβ )) = u(ml ) × · · · × u(m1 ).

Thus there is a canonical bijection: Sr+1 /Sn Drapn (E) (resp. Sr+1 /Sn Drap(E)), n∈typ(E) defined by w → w(Dn ). The relation of inclusion between flags corresponds to the opposite of the relation of inclusion between classes. 2. There is a canonical bijection between the types of relative position between a flag of type m and another of type m and a set of double classes Sm \SE /Sn Relpos(m,n) (E) . As a particular case of 1. It follows that Drapr (E) is a principal homogeneous set under SE .