Ruled cubic surfaces in PG(4,q), Baer subplanes of PG(2,q2) and Hermitian curves

Abstract

In PG(2,q2) let ℓ∞ denote a fixed line, then the Baer subplanes which intersect ℓ∞ in q+1 points are called affine Baer subplanes. Call a Baer subplane of PG(2,q2) non-affine if it intersects ℓ∞ in a unique point. It is shown by Vincenti (Boll. Un. Mat. Ital. Suppl. 2 (1980) 31) and Bose et al. (Utilitas Math. 17 (1980) 65) that non-affine Baer subplanes of PG(2,q2) are represented by certain ruled cubic surfaces in the André/Bruck and Bose representation of PG(2,q2) in PG(4,q) (Math. Z. 60 (1954) 156; J. Algebra 1 (1964) 85; J. Algebra 4 (1966) 117). The André/Bruck and Bose representation of PG(2,q2) involves a regular spread in PG(3,q). For a fixed regular spread View the MathML source, it is known that not all ruled cubic surfaces in PG(4,q) correspond to non-affine Baer subplanes of PG(2,q2) in this manner. In this paper, we prove a characterisation of ruled cubic surfaces in PG(4,q) which represent non-affine Baer subplanes of the Desarguesian plane PG(2,q2). The characterisation relies on the ruled cubic surfaces satisfying a certain geometric condition. This result and the corollaries obtained are then applied to give a geometric proof of the result of Metsch (London Mathematical Society Lecture Note Series, Vol. 245, Cambridge University Press, Cambridge, 1997, p. 77) regarding Hermitian unitals; a result which was originally proved in a coordinate setting