Characterizations of Veronese and Segre varieties

We survey the known and recent characterizations of Segre varieties and Veronesea varieties

Veronesean representations of projective spaces over quadratic associative division algebras

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A characterization of the finite Veronesean by intersection properties

A combinatorial characterization of the Veronese variety of all quadrics in PG(n, q) by means of its intersection properties with respect to subspaces is obtained. The result relies on a similar combinatorial result on the Veronesean of all conics in the plane PG(2, q) by Ferri, Hirschfeld and Thas, and Thas and Van Maldeghem, and a structural characterization of the quadric Veronesean by Thas and Van Maldeghem

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 four codes with automorphism group P Sigma L(3,4) and pseudo-embeddings of the large Witt designs

A pseudo-embedding of a point-line geometry is a representation of the geometry into a projective space over the field F-2 such that every line corresponds to a frame of a subspace. Such a representation is called homogeneous if every automorphism of the geometry lifts to an automorphism of the projective space. In this paper, we determine all homogeneous pseudo-embeddings of the three Witt designs that arise by extending the projective plane PG(2, 4). Along our way, we come across some codes with automorphism group P Sigma L(3, 4) and sets of points of PG(2, 4) that have a particular intersection pattern with Baer subplanes or hyperovals