A uniform characterisation of the varieties of the second row of the Freudenthal–Tits magic square over arbitrary fields
Anneleen De Schepper
Ghent University, BelgiumJeroen Schillewaert
The University of Auckland, New ZealandHendrik van Maldeghem
Ghent University, Belgium
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Abstract
We characterise the projective varieties related to the second row of the Freudenthal–Tits magic square, for both the split and the non-split form, using a common, simple and short geometric axiom system. A special case of our result simultaneously captures the analogues over arbitrary fields of the Severi varieties (comprising the -dimensional module and some of its subvarieties), as well as the Veronese representations of projective planes over composition division algebras (most notably the Cayley plane). It is the culmination of almost four decades of work since the original 1984 result by Mazzocca and Melone who characterised the quadric Veronese variety over a finite field of odd order. The latter result is a finite counterpart to the characterisation of the complex quadric Veronese surface by Severi from 1901.
Cite this article
Anneleen De Schepper, Jeroen Schillewaert, Hendrik van Maldeghem, A uniform characterisation of the varieties of the second row of the Freudenthal–Tits magic square over arbitrary fields. J. Comb. Algebra 7 (2023), no. 3/4, pp. 227–282
DOI 10.4171/JCA/75