Asymmetric linear correlations of projective planes over fields

Projective planes over quadratic 2-dimensional algebras

The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [5, 20, 21]. The geometries appearing in the second row are Severi varieties [24]. We provide an easy uniform axiomatization of these geometries and related ones, over an arbitrary field. In particular we investigate the entry A2 × A2 in the magi...

Finite linear spaces and projective planes

In 1948, De Bruijn and Erdös proved that a finite linear space on v points has at least v lines, with equality occurring if and only if the space is either a near-pencil (all points but one collinear) or a projective plane . In this paper, we study finite linear spaces which are not near-pencils . We obtain a lower bound for the number of lines (as a function of the number of points) for such l...

Homology of Projective Space over Finite Fields

Projective Planes of Type

During the last few years, there has been a mild surge of interest in projective planes of type I-4. This paper is intended to both provide a survey of some recent theorems, and to present some new results concerning these planes. In Section 2, we describe the algebraic setting for these planes. Special attention is given to the structure of the relevant homology groups. In Section 3, some rece...

On projective linear groups over finite fields as Galois groups over the rational numbers

Ideas and techniques from Khare’s and Wintenberger’s article on the proof of Serre’s conjecture for odd conductors are used to establish that for a fixed prime l infinitely many of the groups PSL2(Flr ) (for r running) occur as Galois groups over the rationals such that the corresponding number fields are unramified outside a set consisting of l, the infinite place and only one other prime.