A (simple) arc in finite projective geometry is a set of points which satisfies, in an intuitive way, a feature of curved figures in continuous geometries. Loosely speaking, they are sets of points that are far from "line-like" in a plane or far from "plane-like" in a three-dimensional space. In this finite setting it is typical to include the number of points in the set in the name, so these simple arcs are called -arcs. An important generalization of -arcs, also referred to as arcs in the literature, is the -arcs.
In a finite projective plane (not necessarily Desarguesian) a set of points such that no three points of are collinear (on a line) is called a . If the plane has order then, however the maximum value of can only be achieved if is even. In a plane of order, a -arc is called an oval and, if is even, a -arc is called a hyperoval.
Every conic in the Desarguesian projective plane PG(2,), i.e., the set of zeros of an irreducible homogeneous quadratic equation, is an oval. A celebrated result of Beniamino Segre states that when is odd, every -arc in PG(2,) is a conic (Segre's theorem). This is one of the pioneering results in finite geometry.
If is even and is a -arc in, then it can be shown via combinatorial arguments that there must exist a unique point in (called the nucleus of) such that the union of and this point is a (+ 2)-arc. Thus, every oval can be uniquely extended to a hyperoval in a finite projective plane of even order.
A -arc which can not be extended to a larger arc is called a complete arc. In the Desarguesian projective planes, PG(2,), no -arc is complete, so they may all be extended to ovals.
In the finite projective space PG with, a set of points such that no points lie in a common hyperplane is called a (spatial) -arc. This definition generalizes the definition of a -arc in a plane (where).
A -arc in a finite projective plane (not necessarily Desarguesian) is a set, of points of such that each line intersects in at most points, and there is at least one line that does intersect in points. A -arc is a -arc and may be referred to as simply an arc if the size is not a concern.
The number of points of a -arc in a projective plane of order is at most . When equality occurs, one calls a maximal arc.
Hyperovals are maximal arcs. Complete arcs need not be maximal arcs.