On $α$-points of $q$-analogs of the Fano plane

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 ≥ 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.