Subsets of P^4 with no four points on a plane

TL;DR

This work resolves the problem of constructing a complete track of size $2q+1$ in $\\mathbb{P}^4$ over $\\mathbb{F}_q$ when the quadratic character of $3$ is negative. The authors build the candidate track from the normal rational curve $\\mathcal{N}$ and a derivative set $\\mathcal{V}$ and prove no plane contains four members, yielding a track of size $2q+1$. The core technique couples incidence arguments with a hyperelliptic curve $\\mathcal{C}$ defined by $3F(u)=v^2$ and the Hasse–Weil bound to guarantee a rational point for $q \\\ge 89$, with computational checks for smaller $q$. The result establishes the largest known lower bound for tracks in $\\mathbb{P}^4$ and underscores a substantial gap to the (quasi-linear) upper bound.

Abstract

We describe a new construction of a subset of P^4 with no four points on a plane over any finite field of order q in which 3 is not a square. This set has size 2q + 1, is maximal with respect to inclusion, and is the largest known such set.

Theorems & Definitions (12)

• Theorem 1.1
• Remark 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Theorem 2.4
• proof
• Theorem 2.5