Claudio Bartolone

Topics in Geometric Algebra over Rings

E. ARTIN (1957), R. BAER (1952) and J.DIEUDONNE (1951) emphasized many times the structural identity between classical projective geometry and linear algebra over a division ring. Then Baer pointed out a possible extension of this structural identity to the case of a ring, generating intense research activity in the area of geometric algebra over rings. In this direction the most significant results appear: Ojanguren and Sridharan’s article on the fundamental theorem of projective geometry over a commutative ring; some theorems due to Klingenberg, Bass and Suslin on the structure of the general linear group over appropriate rings and some valuable notes of O’Meara’s on the automorphisms of linear groups over an integral domain.

Algebraic (2, 2)-transformation groups

Abstract In this paper we determine all algebraic transformation groups G, defined over an algebraically closed field k, which operate transitively, but not primitively, on a variety Ω, subject to the following conditions. We require that the (non-effective) action of G on the variety of blocks is sharply 2-transitive, as well as the action on a block Δ of the normalizer G Δ. Also we require sharp transitivity on pairs (X, Y) of independent points of Ω, i.e. points contained in different blocks.

Imprimitive groups highly transitive on blocks

Abstract We classify imprimitive groups acting highly transitively on blocks and satisfying conditions common in geometry. They can be realized as subgroups of twisted wreath products.

A Class of Imprimitive Groups

We classify imprimitive groups inducing the alternating group A4 on the set of blocks, with the inertia subgroup satisfying some very natural geometrical conditions which force the group to operate linearly.

On some Translation Planes Admitting a Frobenius Group of Collineations

Publisher Summary This chapter presents some results concerning translation planes of dimension 2 over GF(q), where q = p r . π denotes such a plane. It is assumed that π has a collineation group F of order q 2 (q-1) satisfying the condition: there exists a point V e l ∞ such that F fixes V and acts (faithfully) as a Frobenius group on l ∞ – {V}.

Le famiglie misurabili di iperboli equilatere del piano

We determine the subgroups of the group of general similarity transformations of the planeR2. This result then allows us to classify the measurable families (following the definition of M. I. Stoka [3]) of non degenerate equilater hyperbolas ofR2. At the same time we give an example of a family of varieties not admitting any measure.

I piani ottenuti “Per rifrazione” da piani sopra quasicorpi associativi ordinati

RiassuntoSi fornisce la nozione diN1-piano e si caratterizzano, algebricamente e geometricamente, gliN1-piani in quanto piani ottenibili da opportuni piani affini ordinati non desarguesiani con una costruzione di “tipo Moulton”.ZusammenfassungUntersucht wird der BegriftN1-Ebenen Sie sind algebraisch und geometrisch dadurch charakterisiert, dass sie mit Hilfe einer Konstruktion vom Typ Moulton von günstigen nicht desarguesschen angeordneten affinen Ebenen erhalten werden können.