On sets without tangents and exterior sets of a conic

TL;DR

This paper explores the structure and construction of sets without tangents and exterior sets of conics in projective planes over finite fields, providing explicit examples, classifications, and extension properties.

Contribution

It introduces new explicit constructions of sets without tangents in PG(2,q), classifies minimal examples in PG(2,5), and analyzes extension properties of exterior conic sets based on field characteristics.

Findings

Smallest sets without tangents in PG(2,5) are of two types.

Explicit construction of larger sets without tangents in PG(2,q).

Extension properties of exterior conic sets depend on q mod 4.

Abstract

A set without tangents in $\PG(2,q)$ is a set of points S such that no line meets S in exactly one point. An exterior set of a conic $\mathcal{C}$ is a set of points $\E$ such that all secant lines of $\E$ are external lines of $\mathcal{C}$. In this paper, we first recall some known examples of sets without tangents and describe them in terms of determined directions of an affine pointset. We show that the smallest sets without tangents in $\PG(2,5)$ are (up to projective equivalence) of two different types. We generalise the non-trivial type by giving an explicit construction of a set without tangents in $\PG(2,q)$, $q=p^h$, $p>2$ prime, of size $q(q-1)/2-r(q+1)/2$, for all $0\leq r\leq (q-5)/2$. After that, a different description of the same set in $\PG(2,5)$, using exterior sets of a conic, is given and we investigate in which ways a set of exterior points on an external line $L$ of a conic in $\PG(2,q)$ can be extended with an extra point $Q$ to a larger exterior set of $\mathcal{C}$. It turns out that if $q=3$ mod 4, $Q$ has to lie on $L$, whereas if $q=1$ mod 4, there is a unique point $Q$ not on $L$.