An algorithm for counting arcs in higher-dimensional projective space

TL;DR

This paper develops a new formula to count n-arcs in higher-dimensional projective spaces, extending Glynn's 2D results to 3D and beyond, with exact counts for small n over finite fields.

Contribution

It introduces a formula for counting n-arcs in projective 3-space and generalizes to higher dimensions, providing explicit counts for small n over finite fields.

Findings

Exact formulas for n-arcs in P^3 over finite fields for n ≤ 7.

Polynomial and quasipolynomial expressions in q for counts.

Extension of Glynn's 2D arc counting formula to higher dimensions.

Abstract

An $n$ arc in $(k-1)$-dimensional projective space is a set of $n$ points so that no $k$ lie on a hyperplane. In 1988, Glynn gave a formula to count $n$-arcs in the projective plane in terms of simpler combinatorial objects called superfigurations. Several authors have used this formula to count $n$-arcs in the projective plane for $n \le 10$. In this paper, we determine a formula to count $n$-arcs in projective 3-space. We then use this formula to give exact expressions for the number of $n$-arcs in $\mathbb{P}^3(\mathbb{F}_q)$ for $n \le 7$, which are polynomial in $q$ for $n \le 6$ and quasipolynomial in $q$ for $n=7$. Lastly, we generalize to higher-dimensional projective space.