Counting arcs in projective planes via Glynn’s algorithm

@article{Kaplan2016CountingAI,
  title={Counting arcs in projective planes via Glynn’s algorithm},
  author={Nathan Kaplan and Susie Kimport and Rachel Lawrence and Luke Peilen and Max Weinreich},
  journal={Journal of Geometry},
  year={2016},
  volume={108},
  pages={1013-1029}
}
An n-arc in a projective plane is a collection of n distinct points in the plane, no three of which lie on a line. Formulas counting the number of n-arcs in any finite projective plane of order q are known for $$n \le 8$$n≤8. In 1995, Iampolskaia, Skorobogatov, and Sorokin counted 9-arcs in the projective plane over a finite field of order q and showed that this count is a quasipolynomial function of q. We present a formula for the number of 9-arcs in any projective plane of order q, even those… 

On the number of $k$-gons in finite projective planes

Let Π be a projective plane of order n and ΓΠ be its Levi graph (the point-line incidence graph). For fixed k ≥ 3, let c2k(ΓΠ) denote the number of 2k-cycles in ΓΠ. In this paper we show that c2k(ΓΠ)

Configurations of noncollinear points in the projective plane

We consider the space $F_n$ of configurations of $n$ points in $P^2$ satisfying the condition that no three of the points lie on a line. For $n = 4, 5, 6$, we compute $H^*(F_n; \mathbb{Q})$ as an

Cohomology of moduli spaces of Del Pezzo surfaces.

We compute the rational Betti cohomology groups of the coarse moduli spaces of geometrically marked Del Pezzo surfaces of degree three and four as representations of the Weyl groups of the

On the cohomology of the space of seven points in general linear position

We determine the cohomology groups of the space of seven points in general linear position in the projective plane as representations of the symmetric group on seven elements by making equivariant

Seven points in general linear position

We determine the cohomology groups of the space of seven points in general linear position as representations of the symmetric group on seven elements by making equivariant point counts over finite

References

SHOWING 1-10 OF 18 REFERENCES

Computational Algebraic Geometry of Projective Configurations

  • B. Sturmfels
  • Mathematics, Computer Science
    J. Symb. Comput.
  • 1991

Open problems in finite projective spaces

Configurations of Points and Lines

This is the only book on the topic of geometric configurations of points and lines. It presents in detail the history of the topic, with its surges and declines since its beginning in 1876. It covers

Linear spaces with at most 12 points

The 28,872,973 linear spaces on 12 points are constructed. The parameters of the geometries play an important role. In order to make generation easy, we construct possible parameter sets for

Combinatorics of Finite Geometries

This textbook is intended as an introduction to the combinatorial theory of finite geometry for undergraduate students of mathematics. Although only a basic knowledge of set theory and analysis is

Practical graph isomorphism, II

The theory of finite linear spaces - combinatorics of points and lines

This paper describes the implementation of n-Dimensional linear spaces in a discrete-time model and some examples show how the model can be modified for flows on rugous topographies varying around an inclined plane.

Rings of geometries I

  • D. Glynn
  • Mathematics
    J. Comb. Theory, Ser. A
  • 1987

Formula for the number of [9, 3] MDS codes

We compute the number of MDS codes of length 9 and dimension 3 over all finite fields, or, what is essentially equivalent, the number of 9-arcs in the projective plane over a finite field.

Dénombrement des configurations dans le plan projectif