On $α$-points of $q$-analogs of the Fano plane
Michael Kiermaier
TL;DR
The paper tackles the existence problem for a $q$-analog of the Fano plane by studying $\\alpha$-points, points whose derived design is the geometric spread. It proves that if a hyperplane consists solely of $\\alpha$-points, then the line set of the symplectic generalized quadrangle $W(q)$ can be partitioned into spreads and the point set of the parabolic quadric $Q(4,q)$ into ovoids, equating two partition phenomena. This leads to a generalized non-$\\alpha$-point blocking-set result: for prime or even $q$, every hyperplane contains a non-$\\alpha$-point, extending known results from $q=2$ to all primes and even $q$. The work links subspace design theory with classical finite geometry, yielding new structural constraints on hypothetical $q$-analogs of the Fano plane and clarifying the geometric consequences of $\\\alpha$-point regularity.
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 $α$-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-$α$-point. For the binary case $q = 2$, Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-$α$-points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of $α$-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 $q$.