291 research outputs found
Symplectic polarities of buildings of type E₆
A symplectic polarity of a building Delta of type E (6) is a polarity whose fixed point structure is a building of type F (4) containing residues isomorphic to symplectic polar spaces. In this paper, we present two characterizations of such polarities among all dualities. Firstly, we prove that, if a duality theta of Delta never maps a point to a neighbouring symp, and maps some element to a non-opposite element, then theta is a symplectic duality. Secondly, we show that, if a duality theta never maps a chamber to an opposite chamber, then it is a symplectic polarity. The latter completes the programme for dualities of buildings of type E (6) of determining all domestic automorphisms of spherical buildings, and it also shows that symplectic polarities are the only polarities in buildings of type E (6) for which the Phan geometry is empty
Semiaffine spaces
In this paper we improve on a result of Beutelspacher, De Vito & Lo Re, who characterized in 1995 finite semiaffine spaces by means of transversals and a condition on weak parallelism. Basically, we show that one can delete that condition completely. Moreover, we extend the result to the infinite case, showing that every plane of a planar space with atleast two planes and such that all planes are semiaffine, comes from a (Desarguesian) projective plane by deleting either a line and all of its points, a line and all but one of its points, a point, or nothing
Characterizations of Veronese and Segre varieties
We survey the known and recent characterizations of Segre varieties and Veronesea varieties
Primitive flag-transitive generalized hexagons and octagons
Suppose that an automorphism group acts flag-transitively on a finite
generalized hexagon or octagon \cS, and suppose that the action on both the
point and line set is primitive. We show that is an almost simple group of
Lie type, that is, the socle of is a simple Chevalley group.Comment: forgot to upload the appendices in version 1, and this is rectified
in version 2. erased cross-ref keys in version 3. Minor revision in version 4
to implement the suggestion by the referee (new section at the end, extended
acknowledgment, simpler proof for Lemma 4.2
Imbrex geometries
We introduce an axiom on strong parapolar spaces of diameter 2, which arises
naturally in the framework of Hjelmslev geometries. This way, we characterize
the Hjelmslev-Moufang plane and its relatives (line Grassmannians, certain
half-spin geometries and Segre geometries). At the same time we provide a more
general framework for a Lemma of Cohen, which is widely used to study parapolar
spaces. As an application, if the geometries are embedded in projective space,
we provide a common characterization of (projections of) Segre varieties, line
Grassmann varieties, half-spin varieties of low rank, and the exceptional
variety by means of a local condition on tangent spaces
Generalized polygons with non-discrete valuation defined by two-dimensional affine R-buildings
In this paper, we show that the building at infinity of a two-dimensional
affine R-building is a generalized polygon endowed with a valuation satisfying
some specific axioms. Specializing to the discrete case of affine buildings,
this solves part of a long standing conjecture about affine buildings of type
G~_2, and it reproves the results obtained mainly by the second author for
types A~_2 and C~_2. The techniques are completely different from the ones
employed in the discrete case, but they are considerably shorter, and general
(i.e., independent of the type of the two-dimensional R-building)
On the varieties of the second row of the split Freudenthal-Tits Magic Square
Our main aim is to provide a uniform geometric characterization of the
analogues over arbitrary fields of the four complex Severi varieties, i.e.~the
quadric Veronese varieties in 5-dimensional projective spaces, the Segre
varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in
14-dimensional projective spaces, and the exceptional varieties of type
in 26-dimensional projective space. Our theorem can be
regarded as a far-reaching generalization of Mazzocca and Melone's approach to
finite quadric Veronesean varieties. This approach takes projective properties
of complex Severi varieties as smooth varieties as axioms.Comment: Small updates, will be published in Annales de l'institut Fourie
Affine twin R-buildings
AbstractIn this paper, we define twinnings for affine R-buildings. We thus extend the theory of simplicial twin buildings of affine type to the non-simplicial case. We show how classical results can be extended to the non-discrete case, and, as an application, we prove that the buildings at infinity of a Moufang twin R-building have the induced structure of a Moufang building. The latter is not true for ordinary “Moufang” R-buildings
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