Nearfield planes and the direction problem in $$AG(2,q^2)$$ A G ( 2 , q 2 )

Abstract

We prove that the projective triangle of $$PG(2,q^2),\,q$$PG(2,q2),q odd, defines via indicator sets a regular nearfield spread of $$PG(3,q)$$PG(3,q), and conversely one of the indicator sets of such a spread is the projective triangle. Then we rephrase our results in the framework of the direction problem. Recall that if $$U$$U is a set of $$s$$s points in $$AG(2,s)$$AG(2,s) and $$N$$N is the number of the determined directions, when $$s=p^2$$s=p2 with $$p$$p an odd prime, Gacs, Lovasz and Sz?nyi have proved that for $$N=\frac{p^2+3}{2}$$N=p2+32 there is a unique example and $$U$$U is affinely equivalent to the graph of the function $$x\mapsto x^{\frac{p^2+1}{2}}$$x?xp2+12. Here we prove a similar result for $$s=q^2,\,q$$s=q2,q any odd prime power, assuming some extra hypotheses.