Slices of the unitary spread

Lunardon, G. ; Parlato, L. ; Pepe, V. ; Trombetti, R.

Journal of Algebraic Combinatorics, Tome 33 (2011) no. 1, pp. 37-56.

Résumé

Summary: We prove that slices of the unitary spread of $Q ^{+}(7, q)$ mathcalQ^+(7,q), $q\equiv 2$ (mod 3), can be partitioned into five disjoint classes. Slices belonging to different classes are non-equivalent under the action of the subgroup of $P\Gamma O ^{+}(8, q)$ fixing the unitary spread. When $q$ is even, there is a connection between spreads of $Q ^{+}(7, q)$ mathcalQ^+(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 $Q ^{+}(7, q)$ mathcalQ^+(7,q), $q=2 ^{2 h+1}$. 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 $P\Gamma O(7, q)$ of the unitary spread of $Q(6, q)$, those among its slices producing spreads of the elliptic quadric $Q ^{ -}(5, q)$ mathcalQ^-(5,q).

Détail

Keywords: keywords ovoid, unitary spread, slice