2-semiarcs in $\mathrm{PG}(2,q)$, $q\leq 13$

TL;DR

This paper classifies 2-semiarcs in projective planes over finite fields for small q, providing complete classifications for q ≤ 7, size spectra for q ≤ 9, and existence results for q=11 and 13.

Contribution

It offers the first complete classification of 2-semiarcs in PG(2,q) for q ≤ 7 and extends the understanding of their sizes and existence for larger q.

Findings

Complete classification for q ≤ 7

Size spectrum determined for q ≤ 9

Existence results for q=11 and q=13

Abstract

A $2$-semiarc is a pointset ${\mathcal S}_k$ with the property that the number of tangent lines to ${\mathcal S}_k$ at each of its points is two. Using some theoretical results and computer aided search, the complete classification of $2$-semiarcs in PG$(2,q)$ is given for $q\leq 7,$ the spectrum of their sizes is determined for $q\leq 9$, and some results about the existence are proven for $q=11$ and $q=13.$ For several sizes of $2$-semiarcs in $\mathrm{PG}(2,q)$, $q\leq 7$, classification results have been obtained by theoretical proofs.