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

Rey Casse, Catherine T. Quinn
2002 Discrete Mathematics
In PG(2; q 2 ) let '∞ denote a ÿxed line, then the Baer subplanes which intersect '∞ in q + 1 points are called a ne Baer subplanes. Call a Baer subplane of PG(2; q 2 ) non-a ne if it intersects '∞ 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-a ne Baer subplanes of PG(2; q 2 ) are represented by certain ruled cubic surfaces in the Andrà e=Bruck and Bose representation of PG(2; q 2 ) in PG(4; q) (Math.
more » ... Z. 60 (1954) 156; J. Algebra 1 (1964) 85; J. Algebra 4 (1966) 117). The Andrà e=Bruck and Bose representation of PG(2; q 2 ) involves a regular spread in PG(3; q). For a ÿxed regular spread S, it is known that not all ruled cubic surfaces in PG(4; q) correspond to non-a ne Baer subplanes of PG(2; q 2 ) in this manner. In this paper, we prove a characterisation of ruled cubic surfaces in PG(4; q) which represent non-a ne Baer subplanes of the Desarguesian plane PG(2; q 2 ). 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.