2012/06/14 by Jeroen Schillewaert, Hendrik Van Maldeghem, Schillewaert, Jeroen +1
Mathematics · #14M15 #17C40 #51A45 #51M35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Differential Geometry (math.DG) #FOS: Mathematics #Meromorphic and Entire Functions #Metric Geometry (math.MG) #math.AC #math.AG #math.DG #math.MG #msc:14M15 #msc:17C40 #msc:51A45 #msc:51M35
paper · pdf · doi:10.48550/arxiv.1206.3021
arxiv created 2012/06/14 · openalex publication_date 2012/06/14 · arxiv updated 2012/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones, over an arbitrary field. In particular we investigate the entry A2 × A2 in the magic square, characterizing Hermitian Veronese varieties, Segre varieties and embeddings of Hjelmslev planes of level 2 over the dual numbers. In fact this amounts to a common characterization of "projective planes over quadratic 2-dimensional algebras", in casu the split and non-split Galois extensions and the dual numbers.