Hyperovals and Unitals in Figueroa Planes

In a finite projective plane of order q, a k-arc is a set of k points no three of which are collinear [9]. A k-arc is complete if it is not contained in a (k+1)-arc. The maximum number of points a k-arc can have is q + 2 if q is even, and q + 1 if q is odd. A (q + 1)-arc is known as an oval. A (q + 2)-arc is known as a hyperoval, in which case every line meets the set in 0 or 2 points. Every oval completes to a hyperoval if q is even. A unital in a projective plane of order q2 is a set U of q3 + 1 points such that every line meets U either in one point or in q+ 1 points. A line is called tangent if it meets the unital in one point, otherwise it is secant. The set of tangent lines to a unital form a unital in the dual plane called the dual unital. See Section 4 for more on unitals. The finite Figueroa planes are a class of projective planes of order q3, with q > 2 a prime power. In the original paper [6], the value of q was restricted, but this restriction was removed in [8]. The description of the planes in [6, 8] was algebraic, but T. Grundhöfer has given a synthetic construction in [7] which is described in Section 2. In [3], W. M. Cherowitzo constructed a class of ovals called the Ovali di Roma in finite Figueroa planes of odd order. In [10], Julia M. Nowlin Brown showed that the Ovali di Roma could be described as the absolute points of a particular polarity of the plane, and that the ovals could be generalized to infinite Figueroa planes of characteristic not two. In the present paper hyperovals are constructed in the finite Figueroa planes of even order (Section 3), and using techniques similar to those in [10] it is shown that all finite Figueroa planes admit a unitary polarity and hence contain unitals (Section 4).