To any spread S of PG(3,q) corresponds a family of locally hermitian ovoids of the Hermitian surface H(3, q^2), and conversely; if in addition S is a semifield spread, then each associated ovoid is a translation ovoid, and conversely. In this paper we calculate the translation group of the locally hermitian ovoids of H(3,q^2) arising from a given semifield spread, and we characterize the p-semiclassical ovoid constructed by Cossidente, Ebert, Marino and Siciliano as the only translation ovoid of H(3,q^2) whose translation group is abelian. If S is a spread of PG(3,q) and O(S) is one of the associated ovoids of H(3,q^2), then using the duality between H(3,q^2) and Q^-(5, q) , another spread of PG(3,q) , say S_1, can be constructed. On the other hand, using the Barlotti-Cofman representation of H(3,q^2), one more spread of a 3-dimensional projective space, say S_2, arises from the ovoid O(S). Lunardon has posed some questions on the relations among S, S_1 and S_2; here we prove that the three spreads are always isomorphic.
Spreads of $PG(3,q)$ and ovoids of polar spaces / Bader, Laura; Marino, G.; Polverino, O.; Trombetti, Rocco. - In: FORUM MATHEMATICUM. - ISSN 0933-7741. - STAMPA. - 19:6(2007), pp. 1101-1110. [10.1515/FORUM.2007.043]
Spreads of $PG(3,q)$ and ovoids of polar spaces
BADER, LAURA;MARINO G.;TROMBETTI, ROCCO
2007
Abstract
To any spread S of PG(3,q) corresponds a family of locally hermitian ovoids of the Hermitian surface H(3, q^2), and conversely; if in addition S is a semifield spread, then each associated ovoid is a translation ovoid, and conversely. In this paper we calculate the translation group of the locally hermitian ovoids of H(3,q^2) arising from a given semifield spread, and we characterize the p-semiclassical ovoid constructed by Cossidente, Ebert, Marino and Siciliano as the only translation ovoid of H(3,q^2) whose translation group is abelian. If S is a spread of PG(3,q) and O(S) is one of the associated ovoids of H(3,q^2), then using the duality between H(3,q^2) and Q^-(5, q) , another spread of PG(3,q) , say S_1, can be constructed. On the other hand, using the Barlotti-Cofman representation of H(3,q^2), one more spread of a 3-dimensional projective space, say S_2, arises from the ovoid O(S). Lunardon has posed some questions on the relations among S, S_1 and S_2; here we prove that the three spreads are always isomorphic.File | Dimensione | Formato | |
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