1,675 research outputs found
An Algorithm for constructing Hjelmslev planes
Projective Hjelmslev planes and Affine Hjelmselv planes are generalisations
of projective planes and affine planes. We present an algorithm for
constructing a projective Hjelmslev planes and affine Hjelsmelv planes using
projective planes, affine planes and orthogonal arrays. We show that all
2-uniform projective Hjelmslev planes, and all 2-uniform affine Hjelsmelv
planes can be constructed in this way. As a corollary it is shown that all
2-uniform Affine Hjelmselv planes are sub-geometries of 2-uniform projective
Hjelmselv planes.Comment: 15 pages. Algebraic Design Theory and Hadamard matrices, 2014,
Springer Proceedings in Mathematics & Statistics 13
Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic Generalisations
We prove the following special case of Mazur's conjecture on the topology of
rational points. Let be an elliptic curve over with
-invariant . For a class of elliptic pencils which are quadratic
twists of by quartic polynomials, the rational points on the projective
line with positive rank fibres are dense in the real topology. This extends
results obtained by Rohrlich and Kuwata-Wang for quadratic and cubic
polynomials.
For the proof, we investigate a highly singular rational curve on the Kummer
surface associated to a product of two elliptic curves over ,
which previously appeared in publications by Mestre, Kuwata-Wang and Satg\'e.
We produce this curve in a simpler manner by finding algebraic equations which
give a direct proof of rationality. We find that the same equations give rise
to rational curves on a class of more general surfaces extending the Kummer
construction. This leads to further applications apart from Mazur's conjecture,
for example the existence of rational points on simultaneous twists of
superelliptic curves.
Finally, we give a proof of Mazur's conjecture for the Kummer surface
without any restrictions on the -invariants of the two elliptic curves.Comment: 14 pages, same content as published version except for added remark
acknowledging overlap with prior work by Ula
Solitons and admissible families of rational curves in twistor spaces
It is well known that twistor constructions can be used to analyse and to
obtain solutions to a wide class of integrable systems. In this article we
express the standard twistor constructions in terms of the concept of an
admissible family of rational curves in certain twistor spaces. Examples of of
such families can be obtained as subfamilies of a simple family of rational
curves using standard operations of algebraic geometry. By examination of
several examples, we give evidence that this construction is the basis of the
construction of many of the most important solitonic and algebraic solutions to
various integrable differential equations of mathematical physics. This is
presented as evidence for a principal that, in some sense, all soliton-like
solutions should be constructable in this way.Comment: 15 pages, Abstract and introduction rewritten to clarify the
objectives of the paper. This is the final version which will appear in
Nonlinearit
String Dualities and Toric Geometry: An Introduction
This note is supposed to be an introduction to those concepts of toric
geometry that are necessary to understand applications in the context of string
and F-theory dualities. The presentation is based on the definition of a toric
variety in terms of homogeneous coordinates, stressing the analogy with
weighted projective spaces. We try to give both intuitive pictures and precise
rules that should enable the reader to work with the concepts presented here.Comment: 17 pages, Latex, 7 figures, invited paper to appear in the special
issue of the Journal of Chaos, Solitons and Fractals on "Superstrings, M, F,
S, ... Theory" (M.S. El Naschie and C. Castro, editors
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