Chapter 4 - Projective Planes

Let Π be a finite projective plane of order n , and let G be a large abelian (or, more generally, quasiregular) collineation group of Π; to be specific, we assume ∣ G ∣ ( n 2 + n + 1)/2. Such planes have been classified into eight cases by Dembowski and Piper in 1967. We survey the present state of knowledge about the existence and structure of such planes. We also discuss some geometric applications, in particular to the construction of arcs and ovals. Technically, a recurrent theme will be the amazing strength of the approach using various types of difference sets and the machinery of integral group rings. Introduction A projective plane is a geometry consisting of points and lines such that any two distinct lines meet in exactly one point, any two distinct points are on exactly one common line, and there are four points no three of which are collinear. Well-known examples are provided by the desarguesian (or classical ) projective planes PG(2, q ), q a prime power, which can be defined over the finite field GF( q ) as follows: points and lines are the 1- and 2-dimensional subspaces of the vector space GF( q ) 3 , and a point is incident with a line if and only if it is a subset of the line. A standard reference for arbitrary projective planes is Hughes and Piper [104]; for the desarguesian case, see Hirschfeld [89].

CHAPTER 1 - BASIC CONCEPTS CONCERNING FINITE GEOMETRIES

CHAPTER 6 - COORDINATIZATION

For , let be a two‐dimensional algebraically defined graph, that is, a bipartite graph where each partite set is a copy of and two vertices and are adjacent if and only if . It is known that has girth 6 and can be extended to the point‐line incidence graph of the classical real projective plane. However, it was unknown whether there exists such that has girth 6 and is nonisomorphic to . This paper answers this question affirmatively and thus provides a construction of a nonclassical real projective plane. This paper also studies the diameter and girth of for families of bivariate functions .

André (1) gave a construction for translation planes from abelian groups possessing “congruences” of subgroups. Schwerdtfeger (3) constructed the plane over a field F from the group of substitutions x → ax + b (a, b ∊ F; a ≠ 0). In this note we describe a construction (inspired by Schwerdtfeger's work), from groups, of planes which are duals of near-field planes.If a plane is (l, m)-transitive (cf. 2, p. 67) for some pair of distinct lines l, m, then the central collineations ϕ with axis m and centre on I may be identified with the “proper” points (that is, points not on I or m) of the plane once an origin O is chosen (not on l or m):

We prove that a non-degenerate homomorphic image of a projective plane is determined to within isomorphism by the inverse image of any one point. An application gives conditions for the preservation of central collineations by a homomorphism.

In a projective plane over a finite field, complete (k,n)-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with k close to the order of the plane, is considered a hard problem. A natural candidate to be a small (k,n)-arc with few characters is the set Ω(C) of the points of a plane curve C of degree n (containing no linear components) such that some line meets C transversally in the plane, i.e. in n pairwise distinct points. Let C be either the Hermitian curve of degree q+1 in PG(2,q2r) with r≥1, or the rational BKS curve of degree q+1 in PG(2,qr) with q odd and r≥1. Then Ω(C) has four and seven characters, respectively. Furthermore, Ω(C) is small as both curves are either maximal or minimal. The completeness problem is investigated by an algebraic approach based on Galois theory and on the Hasse-Weil lower bound. Our main result for the Hermitian case is that Ω(C) is complete for r≥4. For the rational BKS curve, Ω(C) is complete if and only if r is even. If r is odd then the uncovered points by the (q+1)-secants to Ω(C) are exactly the points in PG(2,q) not lying in Ω(C). Adding those points to Ω(C) produces a complete (k,q+1)-arc in PG(2,qr), with k=qr+q. The above results do not hold true for r=2 and there remain open the case r=3 for the Hermitian curve, and the cases r=3,4 for the rational BKS curve. As a by product we also obtain two results of interest in the study of the Galois inverse problem for PGL(2,q).

An elliptic curve can be defined as a smooth projective curve of degree 3 in the projective plane with a distinguished point chosen on it. The set of points on the curve can thus be endowed with a natural additive group structure. The most concrete description of an elliptic curve comes from its affine equation, written as $$y^2=x^3+ax+b,\quad \text{with}\ 4a^3+27b^2\not=0.$$ The theory of elliptic curves is a marvelous mixture of elementary mathematics and profound, advanced mathematics, a mixture which moreover lies on the crossroads of multiple theories: arithmetic, algebraic geometry, group representations, complex analysis, etc. Here, we will provide an introduction to the subject and prove the main Diophantine theorems: the group of rational points is finitely generated (the Mordell-Weil theorem) and the set of integral points is finite (Siegel’s theorem). Finally, we will evoke the famous theorem of Wiles—whose proof resulted in the proof of Fermat’s last theorem—and the Birch & Swinnerton-Dyer conjecture.

Some Sfxf-Blocking Block Designs

On edges crossing few other edges in simple topological complete graphs

We prove that any compact complex surface with c 1 > 0 c_1>0 admits an Einstein metric which is conformally related to a Kähler metric. The key new ingredient is the existence of such a metric on the blow-up C P 2 # 2 C P 2 ¯ \mathbb {CP}_2\# 2\overline {\mathbb {CP}_2} of the complex projective plane at two distinct points.

Let $\mathcal{K}$ denote a nonsingular conic in the complex projective plane. Pascal's theorem says that, given six distinct points $A,B,C,D,E,F$ on $\mathcal{K}$, the three intersection points $AE \cap BF, AD \cap CF, BD \cap CE$ are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal's hexagrammum mysticum.

A two-dimensional G-space,1 in which the geodesic through two distinct points is unique, is either homeomorphic to the plane 2 and all geodesies are isometric to a straight line, or it is homeomorphic to the projective plane P2 and all geodesies are isometric to the same circle, see [l, 10 and 31 ].Two problems arise in either case: (1) To determine the systems of curves (in E2 or P2) which occur as geodesies.( 2) If the geodesies are (or lie on) ordinary straight lines, can the space be imbedded in a higher-dimensional space with the ordinary straight lines as geodesies?The author solved both these problems for E2, [l, Theorems

In the study of finite (Vassiliev’s) knot invariants, Vogel introduced the so-called universal parameters, belonging to the projective plane, in particular parametrizing the simple Lie algebras by Vogel’s table. Subsequently, a number of quantities, such as some universal knot invariants and (quantum) dimensions of simple Lie algebras, have been represented in terms of these parameters, i.e., in the universal form. We prove that at the points from Vogel’s table all known universal quantum dimension formulae are linearly resolvable, i.e., yield finite answers even if these points are singular, provided one restricts them to the appropriate lines. We show that the same phenomenon takes place for another three distinguished points in Vogel’s plane — [Formula: see text], [Formula: see text] and [Formula: see text]. We also examine the same formulae on linear resolvability at the remaining 48 distinguished points in Vogel’s plane, which correspond to the so-called [Formula: see text]-objects. Among them, three points were found to be regular for all known quantum dimension formulae. Two of them happen to be sharing a remarkable similarity with the simple Lie algebras, namely, the universal formulae yield integer-valued outputs (dimensions) at the indicated points in the classical limit.

Screening for mitochondrial DNA (mtDNA) point mutations using nonradioactive single strand conformation polymorphism (SSCP) analysis