Maximal arcs in the complex projective plane and completely 2-valent functions (Q1904261)

An arc in a projective plane is a set of points, no three of which are collinear. The arc is maximal if it is not contained in any arc with more points. An oval is an arc such that there is exactly one tangent at each point. A topological oval (in the real projective plane or the complex projective plane) is an oval that is a closed set of points. Some ``nice'' arcs in the real plane: (1) Topological ovals (2) A convex closed curve that is differentiable at every point but one (3) An arc homeomorphic to the closed unit interval. There are two points \(P\) and \(Q\) through which there are no tangents. Every other point is on an infinite member of tangent. In the complex situation, let \({\mathbb{C}}\) be the set of complex numbers represented by what is usually called the complex plane (actually the real affine plane). Let \(D\) denote the subset of \({\mathbb{C}}\) consisting of the origin, all points on the positive real axis and all points on the upper half plane. Let \(f(z)=z^3\) then the image of \(D\) under \(f\) together with a point \(p_\infty\) is a maximal arc in the complex projective plane. Other examples involve other powers of \(z\).