A Classification of Hyperfocused 12-Arcs

TL;DR

This paper classifies hyperfocused 12-arcs in projective planes over finite fields, linking their existence to specific field sizes and hyperconic structures, and analyzes all 1-factorizations of K12 for embeddability.

Contribution

It provides a complete classification of hyperfocused 12-arcs in PG(2,q), identifying the exact field sizes and geometric configurations where they occur.

Findings

Hyperfocused 12-arcs exist only when q=2^{5k}.

Such arcs are subsets of hyperconics including the nucleus.

The study enumerates and classifies all 1-factorizations of K12 for embeddability.

Abstract

A $k$-arc in PG($2,q$) is a set of $k$ points no three of which are collinear. A hyperfocused $k$-arc is a $k$-arc in which the $k \choose 2$ secants meet some external line in exactly $k-1$ points. Hyperfocused $k$-arcs can be viewed as 1-factorizations of the complete graph $K_k$ that embed in PG($2,q$). We study the 526,915,620 1-factorizations of $K_{12}$, determine which are embeddable in PG($2,q$), and classify hyperfocused $12$-arcs. Specifically we show if a $12$-arc $\mathcal{K}$ is a hyperfocused arc in PG($2,q$) then $q = 2^{5k}$ and $\mathcal{K}$ is a subset of a hyperconic including the nucleus.