Characterizations of Segre varieties

This paper contains strong new characterisations of Segre varieties in finite projective space

Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties

An embedding of a point-line geometry \Gamma is usually defined as an injective mapping \epsilon from the point-set of \Gamma to the set of points of a projective space such that \epsilon(l) is a projective line for every line l of \Gamma, but different situations have lately been considered in the literature, where \epsilon(l) is allowed to be a subline of a projective line or a curve. In this paper we propose a more general definition of embedding which includes all the above situations and we focus on a class of embeddings, which we call Grassmman embeddings, where the points of \Gamma are firstly associated to lines of a projective geometry PG(V), next they are mapped onto points of PG(V\wedge V) via the usual projective embedding of the line-grassmannian of PG(V) in PG(V\wedge V). In the central part of our paper we study sets of points of PG(V\wedge V) corresponding to lines of PG(V) totally singular for a given pseudoquadratic form of V. Finally, we apply the results obtained in that part to the investigation of Grassmann embeddings of several generalized quadrangles

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

Codes and caps from orthogonal Grassmannians

In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding $\varepsilon_k^{gr}$ of an orthogonal Grassmannian $\Delta_k$. In particular, we determine some of the parameters of the codes arising from the projective system determined by $\varepsilon_k^{gr}(\Delta_k)$. We also study special sets of points of $\Delta_k$ which are met by any line of $\Delta_k$ in at most 2 points and we show that their image under the Grassmann embedding $\varepsilon_k^{gr}$ is a projective cap.Comment: Keywords: Polar Grassmannian; dual polar space; embedding; error correcting code; cap; Hadamard matrix; Sylvester construction (this is a slightly revised version of v2, with updated bibliography

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

Some characterizations of finite Hermitian Veroneseans

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