Imbrex geometries

We introduce an axiom on strong parapolar spaces of diameter 2, which arises naturally in the framework of Hjelmslev geometries. This way, we characterize the Hjelmslev-Moufang plane and its relatives (line Grassmannians, certain half-spin geometries and Segre geometries). At the same time we provide a more general framework for a Lemma of Cohen, which is widely used to study parapolar spaces. As an application, if the geometries are embedded in projective space, we provide a common characterization of (projections of) Segre varieties, line Grassmann varieties, half-spin varieties of low rank, and the exceptional variety $\mathcal{E}_{6,1}$ by means of a local condition on tangent spaces

Generalised dual arcs and Veronesean surfaces, with applications to cryptography

We start by defining generalised dual arcs, the motivation for defining them comes from cryptography, since they can serve as a tool to construct authentication codes and secret sharing schemes. We extend the characterisation of the tangent planes of the Veronesean surface V-2(4) in PG(5,q), q odd, described in [J.W.P. Hirschfeld, J.A. Thas, General Galois Geometries, Oxford Math. Monogr., Clarendon Press/Oxford Univ. Press, New York, 1991], as a set of q(2) + q + 1 planes in PG(5, q), such that every two intersect in a point and every three are skew. We show that a set of q 2 + q planes generating PG(5, q), q odd, and satisfying the above properties can be extended to a set of q2 + q + I planes still satisfying all conditions. This result is a natural generalisation of the fact that a q-arc in PG(2, q), q odd, can always be extended to a (q + 1)-arc. This extension result is then used to study a regular generalised dual arc with parameters (9, 5, 2, 0) in PG(9, q), q odd, where we obtain an algebraic characterisation of such an object as being the image of a cubic Veronesean

On the varieties of the second row of the split Freudenthal-Tits Magic Square

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the exceptional varieties of type $\mathsf{E}_{6}$ in 26-dimensional projective space. Our theorem can be regarded as a far-reaching generalization of Mazzocca and Melone's approach to finite quadric Veronesean varieties. This approach takes projective properties of complex Severi varieties as smooth varieties as axioms.Comment: Small updates, will be published in Annales de l'institut Fourie

On exceptional homogeneous compact geometries of type C3

We provide a uniform framework to study the exceptional homogeneous compact geometries of type C-3. This framework is then used to show that these are simply connected, answering a question by Kramer and Lytchak, and to calculate the full automorphism groups

Generalised Veroneseans

In \cite{ThasHVM}, a characterization of the finite quadric Veronesean $\mathcal{V}_{n}^{2^{n}}$ by means of properties of the set of its tangent spaces is proved. These tangent spaces form a {\em regular generalised dual arc}. We prove an extension result for regular generalised dual arcs. To motivate our research, we show how they are used to construct a large class of secret sharing schemes