Symplectic spreads in PG(3, q), inversive planes and projective planes

Abstract

A (line) spread in PG(3, q) is any set of q 2 + 1 disjoint lines in PG(3, q). The spread S is called symplectic if all lines of S are totally isotropic for some symplectic polarity ς of PG(3, q). An ovoid of PG(3, q), q > 2, is a set of q 2 + 1 points, no three of which are collinear; an ovoid of PG(3,2) is the same as an elliptic quadric. An ovoid of the nonsingular quadric Q(4, q) of PG(4, q) is any set O of points of Q(4, q) which has exactly one point in common with each line of Q(4, q). Symplectic spreads of PG(3, q) and ovoids of Q(4, q) are equivalent objects, and, for q even, also ovoids of Q(4, q) and ovoids of PG(3, q) are equivalent objects. With each symplectic spread of PG(3, q) there corresponds a plane of order q 2. A 3 − ( q 2 + 1, q + 1, 1) design is called an inversive plane of order q. With each ovoid of PG(3, q) there corresponds an inversive plane of order q. Here we survey some important characterizations of finite inversive planes, the inversive planes of small order, the planes of small order arising from symplectic spreads of PG(3, q), and all known classes of ovoids of PG(3, q) and Q(4, q). In Appendix A we also survey the ovoids of the nonsingular quadric Q(2 n, q), n ⩾ 3, of PG(2 n, q). In second Appendix B we discuss an interesting relationship between flocks of the quadratic cone K of PG(3, q) and ovoids of Q(4, q).