Hemisystems of Q(6,q), q odd

A family of hemisystems on the parabolic quadrics

“…A subgroup isomorphic to A 5 is then found by computer in the pointwise stabiliser of the conic which fixes a hemisystem. It is not known if the construction in [8] generalises to all q, or to greater rank, and so the authors describe it as sporadic. Our construction instead finds a subgroup in the pointwise stabiliser of the perp of the conic which is isomorphic to Ω 3 (3) ∼ = A 4 , in the case q = 3, and generalises to Ω 3 (q) for all odd q and rank at least 2.…”

“…Moreover, an s+1 2 -ovoid in DQ(2d, q), DH(2d − 1, q 2 ), or DW(2d − 1, q) is a hemisystem of Q(2d, q), H(2d − 1, q 2 ), or W(2d − 1, q), respectively. Recently, Cossidente and Pavese found an infinite family of hemisystems of Q(6, q), q odd, admitting PSL 2 (q 2 ) [8]. This is currently the only known family of hemisystems of the parabolic quadrics, for d 3.…”