On the number of Mordell-Weil generators for cubic surfaces

Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal cardinality which generates S(K) via successive secant and tangent constructions. Let r(S,K) be the cardinality of such a Mordell-Weil generating set. Manin posed what is known as the Mordell-Weil problem for cubic surfaces: if K is finitely generated over its prime subfield then r(S,K) is finite. In this paper, we prove a special case of this conjecture. Namely, if S contains two skew lines both defined over K then r(S,K) = 1. One of the difficulties in studying the secant and tangent process on cubic surfaces is that it does not lead to an associative binary operation as in the case of elliptic curves. As a partial remedy we introduce an abelian group H_S(K) associated to a cubic surface S/K, naturally generated by the K-rational points, which retains much information about the secant and tangent process. In particular, r(S, K) is large as soon as the minimal number of generators of H_S(K) is large. In situations where weak approximation holds, H_S has nice local-to-global properties. We use these to construct a family of smooth cubic surfaces over the rationals such that r(S,K) is unbounded in this family. This is the cubic surface analogue of the unboundedness of ranks conjecture for elliptic curves

The Steiner-Lehmus theorem and "triangles with congruent medians are isosceles" hold in weak geometries

We prove that (i) a generalization of the Steiner-Lehmus theorem due to A. Henderson holds in Bachmann's standard ordered metric planes, (ii) that a variant of Steiner-Lehmus holds in all metric planes, and (iii) that the fact that a triangle with two congruent medians is isosceles holds in Hjelmslev planes without double incidences of characteristic $\neq 3$

Coding Theory and Algebraic Combinatorics

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201

Note di Matematica 26

Abstract. We point out the geometric significance of a part of the theorem regarding the maximality of the orthogonal group in the equiaffine group proved in Keywords: Erlanger Programm, definability, Lω 1 ω -logic MSC 2000 classification: 03C40, 14L35, 51F25, 51A99 A. Schleiermacher and K. Strambach [12] proved a very interesting result regarding the maximaility of the group of orthogonal transformations and of that of Euclidean similarities inside certain groups of affine transformations. Although similar results have been proved earlier, this is the first time that the base field for the groups in question was not the field of real numbers, but an arbitrary Pythagorean field which admits only Archimedean orderings. They also state, as geometric significance of the result regarding the maximality of the group of Euclidean motions in the unimodular group over the reals, that there is "no geometry between the classical Euclidean and the affine geometry". The aim of this note is to point out the exact geometric meaning of the positive part of the 2-dimensional part their theorem, in the case in which the underlying field is an Archimedean ordered Euclidean field. In this case their theorem states that: (1) the group G 1 of Euclidean isometries is maximal in the group H 1 of equiaffinities (affine transformations that preserve non-directed area), and that (2) the group G 2 of Euclidean similarities is maximal in the group H 2 of affine transformations. The restriction to the 2-dimensional case is not essential but simplifies the presentation. The geometric counterpart of group-theoretic results in the spirit of the Erlanger Programm is given by Beth's theorem, as was emphasized by Büch

Point-free foundation of geometry looking at laboratory activities

Researches in "point-free geometry", aiming to found geometry without using points as primitive entities, have always paid attention only to the logical aspects. In this paper, we propose a point-free axiomatization of geometry taking into account not only the logical value of this approach but also, for the first time, its educational potentialities. We introduce primitive entities and axioms, as a sort of theoretical guise that is grafted onto intuition, looking at the educational value of the deriving theory. In our approach the notions of convexity and half-planes play a crucial role. Indeed, starting from the Boolean algebra of regular closed subsets of ℝn, representing, in an excellent natural way, the idea of region, we introduce an n-dimensional prototype of point-free geometry by using the primitive notion of convexity. This enable us to define Re-half-planes, Re-lines, Re-points, polygons, and to introduce axioms making not only meaningful all the given definitions but also providing adequate tools from a didactic point of view. The result is a theory, or a seed of theory, suitable to improve the teaching and the learning of geometry

On Constructive Axiomatic Method

In this last version of the paper one may find a critical overview of some recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure