Abstract
We extend Tits' characterization of the translation ovoids in PG(n, F) to the case that F is a skew field, and give examples. We show that, for n≥3, a translation ovoid Ω in PG(n, F) is (p, q) transitive for some (every) {p, q}⊂Ω⇔F is communtative and Ω is a hyperquadric.
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References
Brauer, R., ‘On a Theorem of H. Cartan’, Bull. Amer. Math. Soc. 55 (1949), 619–620.
Dembowski, P., Finite Geometries, Springer, Berlin, Heidelberg, New York, 1968.
Hartmann, E., ‘Über Moufang-Ovale’, Aequat. Math. 23 (1981), 188–196.
Lam, T. Y., The Algebraic Theory of Quadratic Forms, Benjamin, New York, 1973.
Mäurer, H., ‘Möbius- und Laguerre-Geometrien über schwach konvexen Semiflächen’, Math. Z. 98 (1967), 355–386.
Mäurer, H., ‘Eine Konstruktionsmethode für Möbius Ebenen des Hering-Typs VII 1’, Geom. Dedicata 22 (1987), 247–250.
Pickert, G., Projektive Ebenen (2nd edn), Springer, Berlin, Heidelberg, New York, 1975.
Tits, J., ‘Ovoïdes à translations’, Rend. Mat. Appl. Ser. V 21, 37–59 (1962).
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Dedicated to Professor Dr Helmut R. Salzmann on the occasion of his 60th birthday
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Yaqub, J.C.D.S. Translation ovoids over skew fields. Geom Dedicata 36, 261–271 (1990). https://doi.org/10.1007/BF00150793
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DOI: https://doi.org/10.1007/BF00150793