Non-isomorphic semipartial geometries |
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Authors: | S De Winter |
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Institution: | 1. Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 289 - S22, 9000, Gent, Belgium
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Abstract: | In De Winter and Thas (Des Codes Cryptogr, 32, 153–166, 2004) a semipartial geometry ${\mathcal{S}(\overline{\mathcal{U})}}$ was constructed from any Buekenhout–Metz unital ${\mathcal{U}}$ in PG(2,q2), and it was shown that, although having the same parameters, ${\mathcal{S}(\overline{\mathcal{U})}\not\cong T_2^*(\mathcal{U})}$ , where ${T_2^*\mathcal{U}}$ is the semipartial geometry arising from the linear representation of ${\mathcal{U}}$ . In this note, we will first briefly overview what is known on the geometry ${\mathcal{S}(\overline{\mathcal{U})}}$ (providing shortened unpublished proofs for most results). Then we answer the following question of G. Ebert affirmatively: “Do non-isomorphic Buekenhout–Metz unitals ${\mathcal{U}_1}$ and ${\mathcal{U}_2}$ yield non-isomorphic semipartial geometries ${\mathcal{S}(\overline{\mathcal{U}}_1)}$ and ${\mathcal{S}(\overline{\mathcal{U}}_2)}$ ?”. |
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