\(k\)-arcs, hyperovals, partial flocks and flocks (Q1924176)

This is a survey of results on the objects of the title. A \(k\)-arc in a projective plane is a set of \(k\) points, no three of which are collinear. In a finite projective plane of order \(q\), an oval is a \(k\)-arc with \(k= q+1\). If \(k= q+2\), the \(k\)-arc is a hyperoval. Hyperovals can exist only if \(n\) is even. In all cases, the projective plane is the Desarguesian projective plane \(PG(2,q)\). All known hyperovals are listed. Let \(k\) be a quadratic cone in \(PG(3,q)\). A partition of the points of \(k\) other than the vertex is a flock. A partial flock is a set of disjoint conics on the non-vertex points of \(k\). The author summarizes known relations between \(k\)-arcs and partial flocks.