320 research outputs found
A geometric characterisation of Desarguesian spreads
We provide a characterisation of -spreads in
that have normal elements in general position. In the same way, we obtain a
geometric characterisation of Desarguesian -spreads in
,
Characterisations of elementary pseudo-caps and good eggs
In this note, we use the theory of Desarguesian spreads to investigate good
eggs. Thas showed that an egg in , odd, with two good
elements is elementary. By a short combinatorial argument, we show that a
similar statement holds for large pseudo-caps, in odd and even characteristic.
As a corollary, this improves and extends the result of Thas, Thas and Van
Maldeghem (2006) where one needs at least 4 good elements of an egg in even
characteristic to obtain the same conclusion. We rephrase this corollary to
obtain a characterisation of the generalised quadrangle of
Tits.
Lavrauw (2005) characterises elementary eggs in odd characteristic as those
good eggs containing a space that contains at least 5 elements of the egg, but
not the good element. We provide an adaptation of this characterisation for
weak eggs in odd and even characteristic. As a corollary, we obtain a direct
geometric proof for the theorem of Lavrauw
Pseudo-ovals in even characteristic and ovoidal Laguerre planes
Pseudo-arcs are the higher dimensional analogues of arcs in a projective
plane: a pseudo-arc is a set of -spaces in
such that any three span the whole space. Pseudo-arcs of
size are called pseudo-ovals, while pseudo-arcs of size are
called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from
applying field reduction to an arc in .
We explain the connection between dual pseudo-ovals and elation Laguerre
planes and show that an elation Laguerre plane is ovoidal if and only if it
arises from an elementary dual pseudo-oval. The main theorem of this paper
shows that a pseudo-(hyper)oval in , where is even and
is prime, such that every element induces a Desarguesian spread, is
elementary. As a corollary, we give a characterisation of certain ovoidal
Laguerre planes in terms of the derived affine planes
Field reduction and linear sets in finite geometry
Based on the simple and well understood concept of subfields in a finite
field, the technique called `field reduction' has proved to be a very useful
and powerful tool in finite geometry. In this paper we elaborate on this
technique. Field reduction for projective and polar spaces is formalized and
the links with Desarguesian spreads and linear sets are explained in detail.
Recent results and some fundamental ques- tions about linear sets and scattered
spaces are studied. The relevance of field reduction is illustrated by
discussing applications to blocking sets and semifields
Subgeometries in the Andr\'e/Bruck-Bose representation
We consider the Andr\'e/Bruck-Bose representation of the projective plane
in . We investigate the representation
of -sublines and -subplanes of
, extending the results for of \cite{BarJack2} and
correcting the general result of \cite{BarJack1}. We characterise the
representation of -sublines tangent to or contained in the
line at infinity, -sublines external to the line at infinity,
-subplanes tangent to and -subplanes secant to
the line at infinity
On some subvarieties of the Grassmann variety
Let be a Desarguesian --spread of , a
-dimensional subspace of and the linear set
consisting of the elements of with non-empty intersection with
. It is known that the Pl\"{u}cker embedding of the elements of is a variety of , say . In this paper, we
describe the image under the Pl\"{u}cker embedding of the elements of
and we show that it is an -dimensional algebraic variety, projection of a
Veronese variety of dimension and degree , and it is a suitable linear
section of .Comment: Keywords: Grassmannian, linear set, Desarguesian spread, Schubert
variet
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