Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Generalized hexagons and BLT-sets
- Orthogonally divergent spreads of Hermitian curves
- Lifts of nuclei in finite projective spaces
- Large minimal blocking sets, strong representative systems, and partial unitals
- The complement of a geometric hyperplane in a generalized polygon is usually connected
- Locally co-Heawood graphs
- A theorem of Parmentier characterizing projective spaces by polarities
- Geometries with diagram (diagram omitted)
- Remarks on finite generalized hexagons and octagons with a point-transitive automorphism group
- Block-transitive t-designs, II: large t
- Generalized Fischer spaces
- Ovoids and windows in finite generalized hexagons
- Flag transitive L.C2 geometries
- On nonics, ovals and codes in Desarguesian planes of even order
- Orbits of arcs in projective spaces
- There exists no (76,21,2,7) strongly regular graph
- Group-arcs of prime power order on cubic curves
- Planar Singer groups with even order multiplier groups
- On a footnote of Tits concerning Dn-geometries
- The structure of the central units of a commutative semifield plane
- Partially sharp subsets of PΓL(n, q)
- Partial ovoids and generalized hexagons
- A census of known flag-transitive extended grids
- Root lattice constructions of ovoids
- Coxeter groups in Coxeter groups
- A local characterization of the graphs of alternating forms
- A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2)
- On some locally 3-transposition graphs
- Coherent configurations derived from quasiregular points in generalized quadrangles
- Veldkamp planes
- The Lyons group has no distance-transitive representation
- Intersection of arcs and normal rational curves in spaces of odd characteristic
- Flocks and partial flocks of the quadratic cone in PG(3, q)
- Some extended generalized hexagons
- Nuclei in finite non-Desarguesian projective planes
Intersection of arcs and normal rational curves in spaces of odd characteristic
Published online by Cambridge University Press: 07 September 2010
- Frontmatter
- Contents
- Preface
- Introduction
- Generalized hexagons and BLT-sets
- Orthogonally divergent spreads of Hermitian curves
- Lifts of nuclei in finite projective spaces
- Large minimal blocking sets, strong representative systems, and partial unitals
- The complement of a geometric hyperplane in a generalized polygon is usually connected
- Locally co-Heawood graphs
- A theorem of Parmentier characterizing projective spaces by polarities
- Geometries with diagram (diagram omitted)
- Remarks on finite generalized hexagons and octagons with a point-transitive automorphism group
- Block-transitive t-designs, II: large t
- Generalized Fischer spaces
- Ovoids and windows in finite generalized hexagons
- Flag transitive L.C2 geometries
- On nonics, ovals and codes in Desarguesian planes of even order
- Orbits of arcs in projective spaces
- There exists no (76,21,2,7) strongly regular graph
- Group-arcs of prime power order on cubic curves
- Planar Singer groups with even order multiplier groups
- On a footnote of Tits concerning Dn-geometries
- The structure of the central units of a commutative semifield plane
- Partially sharp subsets of PΓL(n, q)
- Partial ovoids and generalized hexagons
- A census of known flag-transitive extended grids
- Root lattice constructions of ovoids
- Coxeter groups in Coxeter groups
- A local characterization of the graphs of alternating forms
- A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2)
- On some locally 3-transposition graphs
- Coherent configurations derived from quasiregular points in generalized quadrangles
- Veldkamp planes
- The Lyons group has no distance-transitive representation
- Intersection of arcs and normal rational curves in spaces of odd characteristic
- Flocks and partial flocks of the quadratic cone in PG(3, q)
- Some extended generalized hexagons
- Nuclei in finite non-Desarguesian projective planes
Summary
Abstract
We study arcs K in PG(n, q), n ≥ 3, q odd, having many points common with a given normal rational curve L. In particular, we show that, if 0.09q + 2.09 ≥ n ≥ 3, q large, then (q + l)/2 is the largest possible number of points of K on L, improving on the bound given in [11], [12], [14]. When |K ∩ L| = (q + l)/2, we show that the points of K ∩ L are invariant under a cyclic linear collineation of order (q ± l)/2. The corresponding questions for q even are discussed in [13].
Introduction
Let Σ = PG(n, q) denote the n-dimensional projective space over the field GF(q). A k-arc in Σ, with k ≥ n + 1, is a set K of k points such that no n + 1 points of K belong to a hyperplane of Σ. A point r of PG(n, q) extends a k-arc K, in PG(n, q), to a (k + l)-arc if and only if K ∪ {r} is a (k + l)-arc. A k-arc K of PG(n, q) is complete if and only if K is not contained in a (k + l)-arc of PG(n, q). Otherwise, K is called incomplete.
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- Information
- Finite Geometries and Combinatorics , pp. 359 - 378Publisher: Cambridge University PressPrint publication year: 1993
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