Manuscripta Mathematica

, Volume 144, Issue 3–4, pp 401–420 | Cite as

Point configurations on the projective line over a finite field

  • Ernst-Ulrich GekelerEmail author


We study n-point configurations in \({\mathbb{P}^1(\mathbb{F}_q)}\) modulo projective equivalence. For n = 4 and 5, a complete classification is given, along with the numbers of such configurations with a given symmetry group. Using Polya’s coloring theorem, we investigate the behavior of the numbers C(n, q) of classes of n-configurations resp. C spec(n, q) of classes with nontrivial symmetry group. Both are described by rational polynomials in q which depend on q modulo \({\lambda(n) = {\rm lcm} \{m \in \mathbb{N} | m \leq n\}}\) .

Mathematics Subject Classifications (2010)

05A15 14N10 11T99 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aigner M.: A Course in Enumeration. GTM 238. Springer, Berlin (2007)Google Scholar
  2. 2.
    Bohn, M.: Punktkonfigurationen über endlichen Körpern. Bachelor’s Thesis, Saarbrücken (2012)Google Scholar
  3. 3.
    Bombieri E., Husemöller D.: Classification and embeddings of surfaces. In: Proceedings of Symposium in Pure Mathematics 29, Algebraic Geometry Arcata 1974. AMS, Providence, pp. 329–420 (1975)Google Scholar
  4. 4.
    Gekeler E.-U.: Automorphe Formen über \({\mathbb{F}_q(T)}\) mit kleinem Führer. Abh. Math. Sem. Univ. Hamburg 55, 111–146 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Harris M.: Algebraic Geometry. GTM 133. Springer, Berlin (1992)CrossRefGoogle Scholar
  6. 6.
    Hartshorne R.: Algebraic Geometry. GTM 52. Springer, Berlin (1977)CrossRefGoogle Scholar
  7. 7.
    Lang S.: Introduction to Modular Forms. Grundlehren der math. Wiss. 222. Springer, Berlin (1976)Google Scholar
  8. 8.
    Mumford, D.: Geometric Invariant Theory. Ergeb. d. Mathematik 34, Springer, Berlin (1965)Google Scholar
  9. 9.
    Polya G.: Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Math. 68, 145–254 (1937)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.FR 6.1 Mathematik, Campus E2 4Universität des SaarlandesSaarbrückenGermany

Personalised recommendations