Semiarcs with long secants
| Autor: | Bence Csajbók |
|---|---|
| Rok vydání: | 2013 |
| Předmět: |
Plane curve
Blocking set 51E20 51E21 Blocking set Collineation group Semioval Fano plane Type (model theory) Computer Science::Computational Geometry Characterization (mathematics) Upper and lower bounds Theoretical Computer Science Combinatorics Affine group Tangent lines to circles FOS: Mathematics Mathematics - Combinatorics Collineation group Discrete Mathematics and Combinatorics Semiarc Symmetric difference Mathematics Discrete mathematics Applied Mathematics Order (ring theory) Semioval Computational Theory and Mathematics Affine group Blocking set Finite plane Affine plane (incidence geometry) Geometry and Topology Combinatorics (math.CO) Projective plane Finite plane |
| Zdroj: | Scopus-Elsevier |
| ISSN: | 1571-0653 |
| Popis: | In a projective plane $\Pi_q$ of order $q$, a non-empty point set ${\cal S}_t$ is a $t$-semiarc if the number of tangent lines to ${\cal S}_t$ at each of its points is $t$. If ${\cal S}_t$ is a $t$-semiarc in $\Pi_q$, $t1$, then $t$-semiarcs with $q+1-t$ collinear points exist only if $t\geq \sqrt{q-1}$. In $\mathrm{PG}(2,q)$ we prove the lower bound $t\geq(q-1)/2$, with equality only if ${\cal S}_t$ is a blocking set of R\'edei type of size $3(q+1)/2$. We call the symmetric difference of two lines, with $t$ further points removed from each line, a $V_t$-configuration. We give conditions ensuring a $t$-semiarc to contain a $V_t$-configuration and give the complete characterization of such $t$-semiarcs in $\mathrm{PG}(2,q)$. Comment: 12 pages |
| Databáze: | OpenAIRE |
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