Buildings and Schubert schemes by Carlos Contou-Carrere

By Carlos Contou-Carrere

The first a part of this publication introduces the Schubert Cells and different types of the overall linear team Gl (k^(r+1)) over a box ok in response to Ehresmann geometric means. tender resolutions for those kinds are built when it comes to Flag Configurations in k^(r+1) given via linear graphs referred to as minimum Galleries. within the moment half, Schubert Schemes, the common Schubert Scheme and their Canonical tender solution, by way of the occurrence relation in a titties relative construction are built for a Reductive staff Scheme as in Grothendieck's SGAIII. it is a subject the place algebra and algebraic geometry, combinatorics, and workforce thought engage in strange and deep ways.

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9 we know that this intersection rank condition may be written in terms of the vanishing of a set of homogeneous polynomials in the Plücker coordinates of Vi and Wj . On the other hand, observe that the product Drap(k E )m × Drap(k E )n is first embedded in a product of grassmannians, and secondly in a product of projective spaces, by taking the product of the corresponding Plücker embeddings. We conclude that Σ(M ) is a k-variety (cf. 10). 13 A more detailed description of the k-variety structure of Σ(M ) (resp.

By the following definitions we introduce the main objects of interest in this chapter. 10 Let (M, D) ∈ Relpos(E) × Drap(k E ) (resp. (M, D) ∈ Relpos(E) × Drap(E)). Define Σ(M ) ⊂ Drap(k E ) × Drap(k E ) as the fiber over M of the quotient mapping, and Σ(M, D) ⊂ Drap(k E ) as the fiber over D of the mapping Σ(M ) −→ Drap(k E ) induced by the second projection. Write Σ(M, D) = Σ(M, k D ). We call Σ(M ) the Universal Schubert cell of type M , and Σ(M, D) the Schubert cell defined by the flag D and the type M .

9 and the following commutative diagram Drapn (E × Drapm (E)/SE N(l+1)×(λ+1) → l,λ∈N ↓ Drapn (k ) × Drapm (k E )/Gl(k E ). 2) E Remark that the oblique arrow is the mapping (D, D ) → M (D, D ), that the horizontal arrow is injective and that the down arrow is surjective. This achieves the proof. 2 Schubert cells and Schubert varieties In what follows we identify the set Relpos(E) of types of relative position of flags of E with its image in the set of matrices with integral coefficients N(l+1)×λ+1) .

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