Secant Distributions of Unitals

Abstract

Let U be a unital embedded in a projective plane \(\Pi \) of order \(q^2\). For \(R\in U\), let \(s_R\) and \(t_R\) be a secant line through R and the tangent line to U at point R, respectively. If the tangent lines to U, passing through the points in \(s_R\cap U\), intersect at a single point on \(t_R\), then \(s_R\) is referred to as a secant line satisfying the desired property. If \(n_i\) of the points of U have exactly \(m_i\) secant lines satisfying the desired property, then

$$\begin{aligned} m_1^{n_1}, m_2^{n_2}, \cdots \end{aligned}$$

is called the secant distribution of U, where \(\sum n_i=q^3+1\), and \(0\le m_i\le q^2\). In this article, we show that collinear pedal sets of a unital U plays an important role in the secant distribution of U. Formulas for secant distributions of unitals having \(0,1,q^2,\) or \(q^2+q\) special points are provided. Statistics regarding to secant distributions of unitals embedded in planes of orders \(q^2\le 25\) are presented. Some open problems related to secant distributions of unitals having specific number of collinear pedal sets are discussed.