Veronese representation of projective Hjelmslev planes over some quadratic alternative algebras

We geometrically characterise the Veronese representations of ring projective planes over algebras which are analogues of the dual numbers, giving rise to projective Hjelmslev planes of level 2 coordinatised over quadratic alternative algebras. These planes are related to affine buildings of relative type Ã_2 and respective absolute type Ã_2, Ã_5 and Ẽ_6

Characterisations and classifications in the theory of parapolar spaces

This thesis in incidence geometry is divided into two parts, which can both be linked to the geometries of the Freudenthal-Tits magic square. The first and main part consists of an axiomatic characterisation of certain plane geometries, defined via the Veronese mapping using degenerate quadratic alternative algebras (over any field) with a radical that is (as a ring) generated by a single element. This extends and complements earlier results of Schillewaert and Van Maldeghem, who considered such geometries over non-degenerate quadratic alternative algebras. The second and smaller part deals with a classification of parapolar spaces exhibiting the feature that the dimensions of intersections of pairs of symplecta cannot take all possible sensible values, with the only further requirement that, if the parapolar spaces have symplecta of rank 2, then they are strong. This part is based on a joint work with Schillewaert, Van Maldeghem and Victoor

Split buildings of type F₄ in buildings of type E₆

A symplectic polarity of a building of type is a polarity whose fixed point structure is a building of type containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type ). In this paper, we show in a geometric way that every building of type contains, up to conjugacy, a unique class of symplectic polarities. We also show that the natural point-line geometry of each split building of type fully embedded in the natural point-line geometry of arises from a symplectic polarity

Veronesean representations of projective spaces over quadratic associative division algebras

A

On exceptional Lie geometries

Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular, many of the exceptional Lie incidence geometries occur. In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected case of some other well-known characterizations

Geometric characterisation of subvarieties of 6() related to the ternions and sextonions

The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety E_6(K) over an arbitrary field K. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions O' over K (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal-Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other "degenerate composition algebras" as the algebras used to construct the square