Nearfield planes and the direction problem in $\mAthrm{aG}(2,q^2)$

Bader, Laura ; Lunardon, Guglielmo

Journal of Algebraic Combinatorics, Tome 42 (2015) no. 2, pp. 497-505.

Résumé

We prove that the projective triangle of $\mathrm{PG}(2,q^2)$, $q$ odd, defines via indicator sets a regular nearfield spread of $\mathrm{PG}(3,q)$, 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 $U$ is a set of $s$ points in $\mathrm{AG}(2,s)$ and $N$ is the number of the determined directions, when $s=p^2$ with $p$ an odd prime, Gács, Lovász and Szonyi have proved that for $N=\frac{p^2+3}{2}$ there is a unique example and $U$ is affinely equivalent to the graph of the function $x\mapsto x^{\frac{p^2+1}{2}}$. Here we prove a similar result for $s=q^2$, $q$ any odd prime power, assuming some extra hypotheses.

Détail

Classification : 51A40, 51E21
Keywords: nearfield planes, indicator sets, projective triangle, Rédei blocking sets, Baer sublines