Projective planes over quadratic 2-dimensional algebras

TL;DR

This paper provides a uniform axiomatization of geometries related to projective planes over quadratic 2-dimensional algebras, connecting algebraic structures with geometric varieties over arbitrary fields.

Contribution

It offers a new axiomatization of these geometries, including Hermitian Veronese, Segre varieties, and Hjelmslev planes, unifying split and non-split cases over any field.

Findings

Characterization of Hermitian Veronese varieties

Description of Segre varieties and Hjelmslev planes over dual numbers

Unified approach to projective planes over quadratic algebras

Abstract

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 \times 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.