3,735 research outputs found
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
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
Tight contact structures and genus one fibered knots
We study contact structures compatible with genus one open book
decompositions with one boundary component. Any monodromy for such an open book
can be written as a product of Dehn twists around dual non-separating curves in
the once-punctured torus. Given such a product, we supply an algorithm to
determine whether the corresponding contact structure is tight or overtwisted.
We rely on Ozsv{\'a}th-Szab{\'o} Heegaard Floer homology in our construction
and, in particular, we completely identify the -spaces with genus one, one
boundary component, pseudo-Anosov open book decompositions. Lastly, we reveal a
new infinite family of hyperbolic three-manifolds with no co-orientable taut
foliations, extending the family discovered in \cite{RSS}.Comment: 30 pages, 10 figures. Added figures, extended result to all
monodromies, and added sections 5 and
- …