Flocks and partial flocks of quadrics: a survey |
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| Institution: | 1. Department of Mathematics, California Institute of Technology, CA 91125, USA;2. Department of Mathematics, University of California, Irvine, CA 92697, USA;1. Department of Materials Science and Engineering, University of Wisconsin-Madison, 1509 University Avenue, Madison, WI 53706, USA;2. Department of Chemistry, SRM Institute of Science and Technology, Kattankulathur, Chennai 603203, India;3. Department of Biological Systems Engineering, University of Wisconsin–Madison, 460 Henry Mall, Madison, WI 53706, USA;1. MTA–ELTE Geometric and Algebraic Combinatorics Research Group, ELTE Eötvös Loránd University, Budapest, Hungary, Department of Geometry, 1117 Budapest, Pázmány P. stny. 1/C, Hungary;2. Dipartimento di Matematica e Fisica, Università degli Studi della Campania “Luigi Vanvitelli”, Viale Lincoln 5, I-81100 Caserta, Italy;3. Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, Monte S. Angelo, I-80126 Napoli, Italy;1. Department of Mathematics and Informatics, University of Perugia, Perugia, Italy;2. ELKH–ELTE Geometric and Algebraic Combinatorics Research Group, ELTE Eötvös Loránd University, Department of Geometry, 1117 Budapest, Pázmány P. stny. 1/C, Hungary;3. Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kongens Lyngby, Denmark |
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| 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. |
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