8 research outputs found

    Imbrex geometries

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    We introduce an axiom on strong parapolar spaces of diameter 2, which arises naturally in the framework of Hjelmslev geometries. This way, we characterize the Hjelmslev-Moufang plane and its relatives (line Grassmannians, certain half-spin geometries and Segre geometries). At the same time we provide a more general framework for a Lemma of Cohen, which is widely used to study parapolar spaces. As an application, if the geometries are embedded in projective space, we provide a common characterization of (projections of) Segre varieties, line Grassmann varieties, half-spin varieties of low rank, and the exceptional variety E6,1\mathcal{E}_{6,1} by means of a local condition on tangent spaces

    On exceptional Lie geometries

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    Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular, many of the exceptional Lie incidence geometries occur. In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected case of some other well-known characterizations

    Characterisations and classifications in the theory of parapolar spaces

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    This thesis in incidence geometry is divided into two parts, which can both be linked to the geometries of the Freudenthal-Tits magic square. The first and main part consists of an axiomatic characterisation of certain plane geometries, defined via the Veronese mapping using degenerate quadratic alternative algebras (over any field) with a radical that is (as a ring) generated by a single element. This extends and complements earlier results of Schillewaert and Van Maldeghem, who considered such geometries over non-degenerate quadratic alternative algebras. The second and smaller part deals with a classification of parapolar spaces exhibiting the feature that the dimensions of intersections of pairs of symplecta cannot take all possible sensible values, with the only further requirement that, if the parapolar spaces have symplecta of rank 2, then they are strong. This part is based on a joint work with Schillewaert, Van Maldeghem and Victoor

    La "stereotomia scientifica" in Amédée François Frézier

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    Cut-stone constructions are made of pre-hewn blocks dry assembled on top of each other. Owing to the formal complexity characteristic of these works, in order to design them it is necessary to have knowledge of the theory of lines, surfaces and their properties, as well as knowledge of the representation methods capable of rendering them on a plane surface. This knowledge set makes stereotomy the science that anticipates, in terms of theory and tools, modern descriptive geometry. These are the reasons for seeking the beginnings of descriptive geometry in stereotomy, that is, the reasons for the transformation of the mason's art of cutting stone into a bona fide science. Frézier's work fits among the last theoretical essays prior to the géométrie descriptive of Gaspard Monge. It is a treaty on solid geometry, devoted to the shape of the bodies, their intersections and the graphical methods necessary to represent them on a plane. In it the author draws up a rigorous theory that puts in place over two centuries of knowledge and experimentation on the subject of cutting stones
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