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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 in Cossidente A., Ebert G., Marino G., Siciliano A.: Shult Sets and Translation Ovoids of the Hermitian Surface. Adv. Geom. 6 (2006), 475–494 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 2 , 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 1 , arises from the ovoid 𝒪(𝒮). In Lunardon G.: Blocking sets and semifields. J. Comb. Theory Ser. A to appear some questions are posed on the relations among 𝒮, 𝒮 1 and 𝒮 2 ; here we prove that 𝒮 and 𝒮 2 are isomorphic and the ovoids 𝒪(𝒮) and 𝒪(𝒮 1 ), corresponding to 𝒮 and 𝒮 1 respectively under the Plücker map, are isomorphic.
Forum Mathematicum – de Gruyter
Published: Nov 20, 2007
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