Hyperbolic Ovals in Finite Planes (original) (raw)
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On two-transitive parabolic ovals
Discrete Mathematics, 2005
We examine the state of knowledge on the following problem. Let be a finite projective plane of odd order n with an oval and let G be a collineation group of fixing. Assume G fixes a point P on and acts 2-transitively on − {P }. The usual basic question is: what can be said about , and G?
Advances in Geometry, 2000
The following result concerning completely regular ovals is proved: Let P be a projective plane of even order and let O be a completely regular oval with nucleus N. Then P is ðN; NÞ-transitive. Combining this result with previous results [13] one obtains: A projective plane of even order admits a completely regular oval if and only if the plane is dual to a symplectic translation plane. Definition 1.1. Let s be a tangent line to O. We say that O is s-regular if, for every pair of distinct points X ; Y A s V P, there is a third point Z A s V P such that, for every point P 0 N, at least one of the lines PX , PY or PZ is secant to O. If O is s-regular for every tangent line s, then O is called completely regular. A translation plane is called symplectic if it is defined by a spread consisting of maximal totally isotropic subspaces with respect to a nondegenerate alternating bilinear form on the underlying vector space. Classical examples of symplectic planes are Brought to you by | Dipartimento Di Economia Pubblica (Universita degli Studi di Roma La Sapienza Biblioteca Ales Authenticated | 172.16.1.226 Download Date | 5/15/12 2:41 PM
Hyperbolic surfaces in ℙ 3 (ℂ)
Proceedings of the American Mathematical Society
We show a class of perturbations X of the Fermat hypersurface such that any holomorphic curve from ℂ into X is degenerate. Applying this result, we give explicit examples of hyperbolic surfaces in ℙ 3 (ℂ) of arbitrary degree d≥22, and of curves of arbitrary degree d≥19 in ℙ 2 (ℂ) with hyperbolic complements.
Designs, codes and cryptography, 2024
We provide classification results for translation generalized quadrangles of order less than or equal to 64, and hence, for all incidence geometries related to them. The results consist of the classification of all pseudo-ovals in PG(3n − 1, 2), for n = 3, 4, and that of the pseudoovals in PG(3n − 1, q), for n = 5, 6, such that one of the associated projective planes is Desarguesian.
2004
For thousands of years there has been the need to construct ovals! Some of the various approaches available to the artist are discussed (pen and string, mechanical devices, etc.). The shapes of several oval pictures are then analysed using the least squares fitting of elliptical and oval models.
On some hyperbolic planes from finite projective planes
International Journal of Mathematics and Mathematical Sciences, 2001
Let Π = (P ,L,I) be a finite projective plane of order n, and let Π = (P ,L ,I ) be a subplane of Π with order m which is not a Baer subplane (i.e., n ≥ m 2 +m). Consider the substructure Π 0 = (P 0 ,L 0 ,I 0 ) with P 0 = P \{X ∈ P | XIl, l ∈ L }, L 0 = L\L , where I 0 stands for the restriction of I to P 0 ×L 0 . It is shown that every Π 0 is a hyperbolic plane, in the sense of Graves, if n ≥ m 2 +m+1+ m 2 + m + 2. Also we give some combinatorial properties of the line classes in Π 0 hyperbolic planes, and some relations between its points and lines.
A brute force computer aided proof of an existence result about extremal hyperbolic surfaces
Geometry at the Frontier, 2021
Extremal compact hyperbolic surfaces contain a packing of discs of the largest possible radius permitted by the topology of the surface. It is well known that arithmetic conditions on the uniformizing group are necessary for the existence of a second extremal packing in the same surface, but constructing explicit examples of this phenomenon is a complicated task. We present a brute force computational procedure that can be used to produce examples in all cases.