Projective planes over 2-dimensional 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 [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 A 2 × A 2 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.

On the varieties of the second row of the split Freudenthal–Tits Magic Square

“…The case d = 3 is "extra special", since we cannot even apply Lemma 4.6. This case will be the most technical of all (but also the case d = 2 is rather technical, see [25]).…”

Veronese Representation of Projective Hjelmslev Planes Over Some Quadratic Alternative Algebras

“…The Veronese variety of the ring projective plane over the dual numbers is related to a (Pappian) Hjelmslev plane of level 2, see [11]. As was proved in [4], these Hjelmslev planes are the spheres of radius 2 in certain affine buildings of type A 2 and can be related to the split form.…”

“…All these produce X by the above algorithm in a unique way. Since a base change boils down to an element of PGL (11,2), this implies that the stabiliser of X in PGL (11,2) has size at least |PΓL(3, 4)|, and since the point-wise stabiliser must be trivial (as X contains the frame I ∪ { * , •, Σ}), we conclude that this stabiliser is isomorphic to PΓL (3,4). Now define Ξ as the family of 3-spaces spanned by the members of Φ, and still denote by M 10 (F 2 ) the pair (XΞ).…”

“…11 Let V be a vertex. Then, firstly, the set{ρ V (Π Y x ) | x ∈ P V } induces a regular spread R V of v-spaces on Y and the (well-defined) map…”

On exceptional Lie geometries

“…This is in fact our main motivation: single out the properties of the parapolar spaces in the magic square. Moreover, these results will be of use in the investigation of the projective varieties associated with the square [12,15,13], in particular for the study of the Lagrangian Grassmannians (third row) and the adjoint varieties (fourth row).…”