A characterization of two classes of locally truncated diagram geometries

Abstract

Let G ¼ ðP; LÞ be a parapolar space which is locally A nÀ1; 3 ðKÞ for some integer n > 6 and K a field. There exists a class D of 2-convex subspaces, each isomorphic to D 5; 5 ðKÞ, such that every symplecton of G is contained in a unique element of D. Let G ¼ ðP; LÞ be a parapolar space which is locally A nÀ1; 4 ðKÞ for n ¼ 7 or an integer n d 9 and some field K.Assume that G satisfies the extra condition called the Weak Hexagon Axiom. Then there exists a class D of 2-convex subspaces, each isomorphic to D 6; 6 ðKÞ, such that every symplecton of G is contained in a unique element of D. In both of the above cases G is the homomorphic image of a truncated building.2000 Mathematics Subject Classification. 51B25, 51E24 P We prove the following:Brought to you by |

Extending locally truncated chamber Systems by sheaves

“…Theorem 5.3 has been suggested to me by the reading of a paper of Onofrei [14]. Onofrei aims to construct a D W5m -extension of a geometry Γ of type (A?,m)2,2> At that stage, she considers a sheaf for Γ', the completion of which is the required /)",",-extension.…”

On a space related to the affine building of type E7

“…In a previous paper [8], we studied parapolar spaces Γ which were locally of type A n−1,4 (with n > 6 and n = 8) and which satisfied:…”

“…Using the fact that Γ inherits from the local structure two classes of maximal singular subspaces, a new family of objects D was constructed. The elements of D are 2-convex subspaces of type D 6,6 ; see [8,Theorem 2]. Second, using a sheaf theoretic argument it was shown that Γ was a homomorphic image of a truncated building; see [8,Theorem 3.b].…”

“…The elements of D are 2-convex subspaces of type D 6,6 ; see [8,Theorem 2]. Second, using a sheaf theoretic argument it was shown that Γ was a homomorphic image of a truncated building; see [8,Theorem 3.b].…”

“…The case with n = 7 which was not considered in [8] is the subject of this work. The difficulty of this case relies on the fact that all the maximal singular subspaces have the same singular rank, hence a partition of the maximal singular subspaces according to dimension is no longer attainable and therefore many of the arguments based on the local properties cannot be used.…”