On $\alpha$-points of $q$-analogs of the Fano plane

TL;DR

This paper investigates the properties of $ ext{q}$-analogs of the Fano plane, focusing on $ ext{alpha}$-points and their implications for the structure of symplectic generalized quadrangles, extending previous results to all primes and even $ ext{q}$.

Contribution

It establishes a link between hyperplanes of $ ext{alpha}$-points and partitions of symplectic quadrangles into spreads, generalizing prior results beyond the binary case.

Findings

Hyperplanes of only $ ext{alpha}$-points imply partitions into spreads.

Generalization of Heden and Sissokho's result to all primes and even $ ext{q}$.

Connection between $ ext{alpha}$-points and geometric spreads in symplectic quadrangles.

Abstract

Arguably, the most important open problem in the theory of $q$-analogs of designs is the question for the existence of a $q$-analog $D$ of the Fano plane. It is undecided for every single prime power value $q \geq 2$. A point $P$ is called an $\alpha$-point of $D$ if the derived design of $D$ in $P$ is a geometric spread. In 1996, Simon Thomas has shown that there must always exist at least one non-$\alpha$-point. For the binary case $q = 2$, Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-$\alpha$-points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of $\alpha$-points implies the existence of a partiton of the symplectic generalized quadrangle $W(q)$ into spreads. As a consequence, the statement of Heden and Sissokho is generalized to all primes $q$ and all even values of $q$.