A note on characterizing Hermitian curves via intersections with Baer subplanes

Abstract

A unital of the projective plane PG(2, q²) is a set of q³+1 points meeting each line in either 1 or q + 1 points. In [S. G. Barwick, C. M. O'Keefe and L. Storme. Unitals which meet Baer subplanes in 1 modulo q points. J. Geom. 68(2000), 16-22], it was shown that such a unital is a Hermitian curve if it intersects each Baer subplane in either 1, q +1 or 2q +1 points. In the present note, we obtain a two-fold generalization of this. We show that an (arbitrary) set of q³+ 1 points of PG(2,q²) is a Hermitian curve if it intersects each Baer subplane in either 1, q + 1 or 2q + 1 points, and that a set of points of PG(2,q²) (with unspecified size) is a Hermitian curve if it intersects each line in either 0, 1 or q +1 points and each Baer subplane in either 1, q +1 or 2q +1 points. © 2019 Utilitas Mathematica Publishing Inc.. All rights reserved.