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

Arguably, the most important open problem in the theory of q -analogs of designs is the question regarding the existence of a q -analog D of the Fano plane. As of today, it remains undecided for every single prime power order q of the base field. 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 always exists a 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 partition 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 .