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Arcs, Caps and Generalisations in a Finite Projective Space

J. W. P. Hirschfeld, J. A. Thas

TL;DR

The survey comprehensively collects definitions, bounds, and constructions for arcs, caps, and their generalisations in finite projective spaces, with a central focus on generalised ovals and generalised ovoids. It clarifies deep connections to algebraic curves, MDS codes, and finite generalized quadrangles, and details regular versus non-regular structures, translation dualities, and Veronese/Kantor-Knuth constructions. The work highlights major open problems (e.g., classification of ovoids in certain spaces, regularity questions for pseudo-ovoids, and kernels in Moufang GQ classifications) and underscores the rich interplay between combinatorial geometry, finite geometry, and group-theoretic methods. Overall, it provides a unifying framework for understanding high-dimensional generalisations and their roles in associated incidence structures and coding theory.

Abstract

Arcs and caps are fundamental structures in finite projective spaces. They can be generalised. Here, a survey is given of some important results on these objects, in particular on generalised ovals and generalised ovoids. The paper also contains recent results and several open problems.

Arcs, Caps and Generalisations in a Finite Projective Space

TL;DR

The survey comprehensively collects definitions, bounds, and constructions for arcs, caps, and their generalisations in finite projective spaces, with a central focus on generalised ovals and generalised ovoids. It clarifies deep connections to algebraic curves, MDS codes, and finite generalized quadrangles, and details regular versus non-regular structures, translation dualities, and Veronese/Kantor-Knuth constructions. The work highlights major open problems (e.g., classification of ovoids in certain spaces, regularity questions for pseudo-ovoids, and kernels in Moufang GQ classifications) and underscores the rich interplay between combinatorial geometry, finite geometry, and group-theoretic methods. Overall, it provides a unifying framework for understanding high-dimensional generalisations and their roles in associated incidence structures and coding theory.

Abstract

Arcs and caps are fundamental structures in finite projective spaces. They can be generalised. Here, a survey is given of some important results on these objects, in particular on generalised ovals and generalised ovoids. The paper also contains recent results and several open problems.

Paper Structure

This paper contains 26 sections, 48 theorems, 17 equations.

Table of Contents

  1. Notation
  2. Introduction
  3. Arcs, ovals and hyperovals in $\mathrm{PG}(2,q)$
  4. Arcs in $\mathrm{PG}(n,q),\ n\geq 3$
  5. Caps and ovoids
  6. Caps and ovoids in $\mathrm{PG}(3,q)$
  7. Caps in $\mathrm{PG}(n,q),\ n\geq 3$
  8. Generalised ovals
  9. Introduction
  10. Generalised $k$-arcs
  11. Pseudo-ovals and pseudo-hyperovals
  12. Regular pseudo-ovals and regular pseudo-hyperovals
  13. Characterisations
  14. Pseudo-ovals, pseudo-hyperovals and spreads
  15. The case $n=2$
  16. ...and 11 more sections

Key Result

Theorem 3.2

(hirJ98) Let ${\cal K}$ be a $k$-arc of $\mathrm{PG}(2,q)$. Then

Theorems & Definitions (99)

  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 3.6
  • Definition 4.1
  • Definition 4.2
  • Theorem 4.3
  • proof
  • ...and 89 more