Flocks and partial flocks of quadrics: a survey

Abstract

Abstract If O is an ovoid of PG(3, q ), then a partition of all but two points of O into q −1 disjoint ovals is called a flock of O . A partition of a nonsingular hyperbolic quadric Q + (3, q ) into q +1 disjoint irreducible conics is called a flock of Q + (3, q ). Further, if O is either an oval or a hyperoval of PG(2, q ) and if K is the cone with vertex a point x of PG(3, q )⧹PG(2, q ) and base O , then a partition of K ⧹{ x } into q disjoint ovals or hyperovals in the respective cases is called a flock of K . The theory of flocks has applications to projective planes, generalized quadrangles, hyperovals, inversive planes; using flocks new translation planes, hyperovals and generalized quadrangles were discovered. Let Q be an elliptic quadric, a hyperbolic quadric or a quadratic cone of PG(3, q ). A partial flock of Q is a set P consisting of β disjoint irreducible conics of Q . Partial flocks which are no flocks, have applications to k -arcs of PG(2, q ), to translation planes and to partial line spreads of PG(3, q ). Recently, the definition and many properties of flocks of quadratic cones in PG(3, q ) were generalized to partial flocks of quadratic cones with vertex a point in PG( n , q ), for n ⩾3 odd.