Projective planes over quadratic 2-dimensional algebras
نویسندگان
چکیده
The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [5, 20, 21]. The geometries appearing in the second row are Severi varieties [24]. 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, the inseparable extensions of degree 2 in characteristic 2 and the dual numbers.
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