75 research outputs found
Veldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties
Let be a Segre variety that is -fold direct product of projective
lines of size three. Given two geometric hyperplanes and of
, let us call the triple the
Veldkamp line of . We shall demonstrate, for the sequence , that the properties of geometric hyperplanes of are fully
encoded in the properties of Veldkamp {\it lines} of . Using this
property, a complete classification of all types of geometric hyperplanes of
is provided. Employing the fact that, for , the
(ordinary part of) Veldkamp space of is , we shall
further describe which types of geometric hyperplanes of lie on a
certain hyperbolic quadric that
contains the and is invariant under its stabilizer group; in the
case we shall also single out those of them that correspond, via the
Lagrangian Grassmannian of type , to the set of 2295 maximal subspaces
of the symplectic polar space .Comment: 16 pages, 8 figures and 7 table
The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings
AbstractWe characterize the hyperplanes of the dual polar space DW(2n−1,q) which arise from projective embeddings as those hyperplanes H of DW(2n−1,q) which satisfy the following property: if Q is an ovoidal quad, then Q∩H is a classical ovoid of Q. A consequence of this is that all hyperplanes of the dual polar spaces DW(2n−1,4), DW(2n−1,16) and DW(2n−1,p) (p prime) arise from projective embeddings
Central aspects of skew translation quadrangles, I
Except for the Hermitian buildings , up to a combination
of duality, translation duality or Payne integration, every known finite
building of type satisfies a set of general synthetic
properties, usually put together in the term "skew translation generalized
quadrangle" (STGQ). In this series of papers, we classify finite skew
translation generalized quadrangles. In the first installment of the series, as
corollaries of the machinery we develop in the present paper, (a) we obtain the
surprising result that any skew translation quadrangle of odd order is
a symplectic quadrangle; (b) we determine all skew translation quadrangles with
distinct elation groups (a problem posed by Payne in a less general setting);
(c) we develop a structure theory for root-elations of skew translation
quadrangles which will also be used in further parts, and which essentially
tells us that a very general class of skew translation quadrangles admits the
theoretical maximal number of root-elations for each member, and hence all
members are "central" (the main property needed to control STGQs, as which will
be shown throughout); (d) we solve the Main Parameter Conjecture for a class of
STGQs containing the class of the previous item, and which conjecturally
coincides with the class of all STGQs.Comment: 66 pages; submitted (December 2013
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