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Hyperbolic unitals in the Hall planes

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Abstract

A new transformation method for incidence structures was introduced in [8],an open problem is to characterize classical incidence structures obtained by transformation of others. In this work we give some, sufficient conditions to transform, with the procedure of [8],a unital embedded in a projective plane into another one. As application of this result we construct unitals in the Hall planes by transformation of the hermitian curves and we give necessary and sufficient conditions for the constructed unitals to be projectively equivalent. This allows to find different classes of not projectively equivalent Buekenhout's unitals, [2],and to find the class of unitals descovered by Grüning, [4],easily proving its embeddability in the dual of a Hall plane. Finally we prove that the affine unital associated to the unital of [4]is isomorphic to the affine hyperbolic hermitian curve.

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Authors and Affiliations

  1. Dipartimento di Matematica, Universitá di Modena, via Campi 213/B, I-41100, Modena, Italy

    Gloria Rinaldi

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  1. Gloria Rinaldi
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Work performed under the auspicies of G.N.S.A.G.A. and supported by 40% grants of M.U.R.S.T.

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Rinaldi, G. Hyperbolic unitals in the Hall planes. J Geom 54, 148–154 (1995). https://doi.org/10.1007/BF01222862

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  • DOI: https://doi.org/10.1007/BF01222862

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