The geometry of null systems, Jordan algebras and von Staudt's theorem

Bertram, Wolfgang

Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 193-225.

We characterize an important class of generalized projective geometries $(X,X^{\text{'}})$ by the following essentially equivalent properties: (1) $(X,X^{\text{'}})$ admits a central null-system; (2) $(X,X^{\text{'}})$ admits inner polarities: (3) $(X,X^{\text{'}})$ is associated to a unital Jordan algebra. These geometries, called of the first kind, play in the category of generalized projective geometries a rôle comparable to the one of the projective line in the category of ordinary projective geometries. In this general set-up, we prove an analogue of von Staudt’s theorem which generalizes similar results by L.K. Hua.

Nous caractérisons une classe importante de géométries projectives généralisées $(X,X^{\text{'}})$ par les propriétés équivalentes suivantes : (1) $(X,X^{\text{'}})$ admet une polarité nulle centrale; (2) $(X,X^{\text{'}})$ admet une polarité intérieure; (3) $(X,X^{\text{'}})$ est associée à une algèbre de Jordan avec élément neutre. Dans ce cadre, nous démontrons un analogue du théorème de von Staudt qui généralise des résultats similaires de L.K. Hua.

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