ABSTRACT. Let P be a finite classical polar space of rank r, r t> 2. An ovoid O of P is a pointset of P, which has exactly one point in common with every totally isotropic subspace of rank r. It is proved that the polar space W,(q) arising from a symplectic polarity of PG(n, q), n odd and n > 3, that the polar space Q(2n, q) arising from a non-singular quadric in PG(2n, q), n > 2 and q even, that the polar space Q-(2n + 1, q) arising from a non-singular elliptic quadric in PG(2n + 1, q), n > 1, and that the polar space H(n, q2) arising from a non-singular Hermitian variety in PG(n, q2), n even and n > 2, have no ovoids.Let S be a generalized hexagon of order n (>/1). If V is a pointset of order n3+ 1 of S, such that every two points are at distance 6, then V is called an ovoid of S. If H(q) is the classical generalized hexagon arising from G2(q), then it is proved that H(q) has an ovoid iff Q(6, q) has an ovoid. There follows that Q(6, q), q = 3 zh+a, has an ovoid, and that H(q), q even, has no ovoid.A regular system of order m on H(3, q2) is a subset K of the lineset of H(3, q2), such that through every point of H(3, q2) there are m (> 0) lines of K. B. Segre shows that, if K exists, then m = q + 1 or (q + 1)/2. If m = (q + 1)/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).
OvoiDs AND SPREADSLet P be a finite classical polar space of rank (or index) r, r >/2 [3]. An ovoid O of P is a pointset of P, which has exactly one point in common with every totally isotropic subspace of rank r. A spread S of P is a set of maximal totally isotropic subspaces, which constitutes a partition of the pointset.We shall use the following notation:
W,(q)the polar space arising from a symplectic polarity of PG(n, q), n odd; Q(2n, q) the polar space arising from a non-singular quadric Q in
PG(2n, q);Q+ (2n + 1, q) the polar space arising from a non-singular hyperbolic quadric Q + [7] in PG(2n + 1, q); Q-(2n + 1, q) the polar space arising from a non-singular elliptic quadric Q- [7] in PG(2n + 1, q); H(n, q2) the polar space arising from a non-singular Hermitian variety H [7] in PG(n, q2).The following results are known:(a) W,(q), n odd, has always a (regular) spread (the proof given in [16] for n = 5, extends to any odd n). W3(q) has an ovoid iff q is even [19]; every ovoid of W3(q), q even, is an ordinary ovoid of PG(3, q) and every ordinary ovoid of PG(3, q), q even, is an ovoid of some W3(q) [18].
