Slices of the unitary spread |
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Authors: | G Lunardon L Parlato V Pepe R Trombetti |
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Institution: | 1.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”,Università degli Studi di Napoli “Federico II”,Napoli,Italy;2.Department of Pure Mathematics and Computer Algebra,University of Gent,Gent,Belgium |
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Abstract: | We prove that slices of the unitary spread of Q+(7,q)\mathcal{Q}^{+}(7,q), q≡2 (mod 3), can be partitioned into five disjoint classes. Slices belonging to different classes are non-equivalent under
the action of the subgroup of PΓO
+(8,q) fixing the unitary spread. When q is even, there is a connection between spreads of Q+(7,q)\mathcal{Q}^{+}(7,q) and symplectic 2-spreads of PG(5,q) (see Dillon, Ph.D. thesis, 1974 and Dye, Ann. Mat. Pura Appl. (4) 114, 173–194, 1977). As a consequence of the above result we determine all the possible non-equivalent symplectic 2-spreads arising from the
unitary spread of Q+(7,q)\mathcal{Q}^{+}(7,q), q=22h+1. Some of these already appeared in Kantor, SIAM J. Algebr. Discrete Methods 3(2), 151–165, 1982. When q=3
h
, we classify, up to the action of the stabilizer in PΓO(7,q) of the unitary spread of Q(6,q), those among its slices producing spreads of the elliptic quadric Q-(5,q)\mathcal{Q}^{-}(5,q). |
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