208 research outputs found
On transparent embeddings of point-line geometries
We introduce the class of transparent embeddings for a point-line geometry
as the class of full projective
embeddings of such that the preimage of any projective
line fully contained in is a line of . We
will then investigate the transparency of Pl\"ucker embeddings of projective
and polar grassmannians and spin embeddings of half-spin geometries and dual
polar spaces of orthogonal type. As an application of our results on
transparency, we will derive several Chow-like theorems for polar grassmannians
and half-spin geometries.Comment: 28 Pages/revised version after revie
Highest weight modules and polarized embeddings of shadow spaces
Let Gamma be the K-shadow space of a spherical building Delta. An embedding V
of Gamma is called polarized if it affords all "singular" hyperplanes of Gamma.
Suppose that Delta is associated to a Chevalley group G. Then Gamma can be
embedded into what we call the Weyl module for G of highest weight lambda_K. It
is proved that this module is polarized and that the associated minimal
polarized embedding is precisely the irreducible G-module of highest weight
lambda_K. In addition a number of general results on polarized embeddings of
shadow spaces are proved. The last few sections are devoted to the study of
specific shadow spaces, notably minuscule weight geometries, polar
grassmannians, and projective flag-grassmannians. The paper is in part
expository in nature so as to make this material accessible to a wide audience.Comment: Improvement in exposition of Sections 1-3 and . Notation improved.
References added. Main results unchange
Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties
These are expository notes from the 2008 Srni Winter School. They have two
purposes: (1) to give a quick introduction to exterior differential systems
(EDS), which is a collection of techniques for determining local existence to
systems of partial differential equations, and (2) to give an exposition of
recent work (joint with C. Robles) on the study of the Fubini-Griffiths-Harris
rigidity of rational homogeneous varieties, which also involves an advance in
the EDS technology.Comment: To appear in the proceedings of the 2008 Srni Winter School on
Geometry and Physic
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
On hyperovals of polar spaces
We derive lower and upper bounds for the size of a hyperoval of a finite polar space of rank 3. We give a computer-free proof for the uniqueness, up to isomorphism, of the hyperoval of size 126 of H(5, 4) and prove that the near hexagon E-3 has up to isomorphism a unique full embedding into the dual polar space DH(5, 4)
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