Abstract
LetP be a finite classical polar space of rankr, r⩾2. An ovoidO ofP is a pointset ofP, which has exactly one point in common with every totally isotropic subspace of rankr. It is proved that the polar spaceW n (q) arising from a symplectic polarity ofPG(n, q), n odd andn > 3, that the polar spaceQ(2n, q) arising from a non-singular quadric inPG(2n, q), n > 2 andq even, that the polar space Q−(2n + 1,q) arising from a non-singular elliptic quadric inPG(2n + 1,q), n > 1, and that the polar spaceH(n,q 2) arising from a non-singular Hermitian variety inPG(n, q 2)n even andn > 2, have no ovoids.
LetS be a generalized hexagon of ordern (⩾1). IfV is a pointset of order n3 + 1 ofS, such that every two points are at distance 6, thenV is called an ovoid ofS. IfH(q) is the classical generalized hexagon arising fromG 2 (q), then it is proved thatH(q) has an ovoid iffQ(6, q) has an ovoid. There follows thatQ(6, q), q=32h+1, has an ovoid, and thatH(q), q even, has no ovoid.
A regular system of orderm onH(3,q 2) is a subsetK of the lineset ofH(3,q 2), such that through every point ofH(3,q 2) there arem (> 0) lines ofK. B. Segre shows that, ifK exists, thenm=q + 1 or (q + l)/2.If m=(q + l)/2,K is called a hemisystem. The last part of the paper gives a very short proof of Segre's result. Finally it is shown how to construct the 4-(11, 5, 1) design out of the hemisystem with 56 lines (q=3).
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Thas, J.A. Ovoids and spreads of finite classical polar spaces. Geom Dedicata 10, 135–143 (1981). https://doi.org/10.1007/BF01447417
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DOI: https://doi.org/10.1007/BF01447417