Unitals which meet Baer subplanes in 1 modulo \(q\) points
Let \(\pi\) be a translation plane of order \(q^2\) with kernel containing \(GF(q)\). A unital in \(\pi\) is a set \(\mathcal U\) of \(q^3+1\) points such that each line meets \(\mathcal U\) in either \(1\) or \(q+1\) points. A unital in \(\pi\) is called parabolic if the translation line of \(\pi\) is a tangent of \(\mathcal U\). The authors prove that a parabolic unital \(\mathcal U\) in \(\pi\) is a Buekenhout-Metz unital (a parabolic unital arising by a construction due to Buekenhout) if and only if suitable Baer subplanes containing the translation line of \(\pi\) meet \(\mathcal U\) in \(1\) modulo \(q\) points. As a by-product, they show that a unital \(\mathcal U\) in \(PG(2,q^2)\) is classical (a Hermitian curve) if and only if it meets each Baer subplane of \(PG(2,q^2)\) in \(1\) modulo \(q\) points.
scientific article; zbMATH DE number 1507991
scientific article