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

Abstract

AbstractArguably, 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 $$\alpha $$ 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-$$\alpha $$ α -point. For the binary case $$q = 2$$ q = 2 , Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-$$\alpha $$ α -points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of $$\alpha $$ 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.