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Non-projective embeddings in the grassmann variety
No full texte investigate properties of the grassmann embedding of dual classical thick generalized quadrangles focusing on the grassmann embedding of the dual of an orthogonal quadrangle and the dual of a hermitian quadrangle We prove that, if the characteristic of the field is different from 2 then the dimension of the grassmann embedding of is and its image is isomorphic to the quadratic veronese variety of a 3-dimensional projective space. If is a perfect field of characteristic 2 then the dimension of the grassmann embedding of is proved to be and its image is a -dimensional algebraic subvariety of the grassmannian of lines of a -dimensional projective space.
Moving to consider the dual quadrangle , we prove that the dimension of its grassmann embedding is and the image of under the grassmann embedding is a -dimensional algebraic subvariety of the grassmannian of lines of a -dimensional projective space
An outline of polar spaces: basics and advances
No full textThis paper is an extended version of a series of lectures on polar spaces given during the workshop and conference \lq Groups and Geometries\rq\, held at the Indian Statistical Institute in Bangalore in December 2012.
We firstly give a concise exposition of the theory of polar spaces, ending up with the classification of polar spaces of rank at least . Then we present a few related research topics, as polar spaces of infinite rank, non-linear embeddings of polar spaces, projective embeddings of dual polar spaces and polar grassmannians
The q-clan geometries with q=2e
No full textLa monografia rappresenta una trattazione coerente e completa della teoria dei quadrangoli generalizzati associati a flock in caratteristica 2 fornendo una descrizione delle principale strutture geometriche ad essi associate e riportandone i risultati piu' significativi
Minimum distance of symplectic Grassmann codes
Full text linkIn this paper we introduce symplectic Grassmann codes, in analogy to ordinary Grassmann codes and orthogonal Grassmann codes, as projective codes defined by symplectic Grassmannians. Lagrangian–Grassmannian codes are a special class of symplectic Grassmann codes. We describe all the parameters of line symplectic Grassmann codes and we provide the full weight enumerator for the Lagrangian–Grassmannian codes of rank 2 and 3
Some results on caps and codes related to orthogonal Grassmannians --- a preview
No full textIn this note we offer a short summary of some recent
results, to be contained in a
forthcoming paper \cite{CG}, on projective caps and linear
error correcting codes arising from the Grassmann embedding
of an orthogonal Grassmannian More precisely, we consider the codes arising from the projective system determined by
and determine some of their parameters.
We also investigate special sets of points of which are met
by any line of in at most points proving that their image under the Grassmann embedding is a projective cap
Grassmann and Weyl embeddings of orthogonal grassmannians
Full text linkGiven a non-singular quadratic form of maximal Witt index on V := V(2n+1,\F), let be the building of type formed by the subspaces of totally singular for and, for , let be the -grassmannian of . Let be the embedding of into \PG(\bigwedge^kV) mapping every point of to the point of \PG(\bigwedge^k V). It is known that if \mathrm{char}(\F)\neq 2 then . In this paper we give a new very easy proof of this fact. We also prove that if \mathrm{char}(\F) = 2 then . As a consequence, when or a number field, and or , then is universal
Two forms related to the symplectic dual polar space in odd characteristic
No full textLet be a -dimensional vector space over a field
equipped with a non-degenerate alternating form
Let be the -grassmannian of \PG(V) and
the dual of the polar space associated to . Then and are
naturally embedded in the vector space and
respectively, where dim()
and dim() The spaces
and can be regarded as modules for the symplectic
group
If char(), we will define two forms
and of which coincide on and we will
investigate the relation between these two forms and the
collineation of naturally induced by . We will obtain a
description of the module in terms of the two subspaces of where
the linear functionals induced by and are equal and respectively opposite.\
The simple connectedness of hyperplane complements in thick dual polar spaces of rank at least 4
No full textLet be a dual polar space of rank , be a hyperplane of
and be the complement of in . We shall prove that, if all lines of have more than points, then is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings
Regular partitions of half-spin geometries
Full text linkWe describe several families of regular partitions of half-spin geometries and determine their associated parameters and eigenvalues. We also give a general method for computing the eigenvalues of regular partitions of half-spin geometries
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