Arc (projective geometry)



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 $k$-arcs. An important generalization of $k$-arcs, also referred to as arcs in the literature, is the ($k, d$)-arcs.

$k$-arcs in a projective plane
In a finite projective plane $\pi$ (not necessarily Desarguesian) a set $A$ of $k (k ≥ 3)$ points such that no three points of $A$ are collinear (on a line) is called a $k - arc$. If the plane π has order $q$ then $k ≤ q + 2$, however the maximum value of $k$ can only be achieved if $q$ is even. In a plane of order $q$, a $(q + 1)$-arc is called an oval and, if $q$ is even, a $(q + 2)$-arc is called a hyperoval.

Every conic in the Desarguesian projective plane PG(2,$q$), i.e., the set of zeros of an irreducible homogeneous quadratic equation, is an oval. A celebrated result of Beniamino Segre states that when $q$ is odd, every $(q + 1)$-arc in PG(2,$q$) is a conic (Segre's theorem). This is one of the pioneering results in finite geometry.

If $q$ is even and $A$ is a $(q + 1)$-arc in π, then it can be shown via combinatorial arguments that there must exist a unique point in π (called the nucleus of $A$) such that the union of $A$ and this point is a ($q$ + 2)-arc. Thus, every oval can be uniquely extended to a hyperoval in a finite projective plane of even order.

A $k$-arc which can not be extended to a larger arc is called a complete arc. In the Desarguesian projective planes, PG(2,$q$), no $q$-arc is complete, so they may all be extended to ovals.

$k$-arcs in a projective space
In the finite projective space PG($n, q$) with $n ≥ 3$, a set $A$ of $k ≥ n + 1$ points such that no $n + 1$ points lie in a common hyperplane is called a (spatial) $k$-arc. This definition generalizes the definition of a $k$-arc in a plane (where $n = 2$).

($k, d$)-arcs in a projective plane
A ($k, d$)-arc ($k, d > 1$) in a finite projective plane π (not necessarily Desarguesian) is a set, $A$ of $k$ points of π such that each line intersects $A$ in at most $d$ points, and there is at least one line that does intersect $A$ in $d$ points. A ($k, 2$)-arc is a $k$-arc and may be referred to as simply an arc if the size is not a concern.

The number of points $k$ of a ($k, d$)-arc $A$ in a projective plane of order $q$ is at most $qd + d − q$. When equality occurs, one calls $A$ a maximal arc.

Hyperovals are maximal arcs. Complete arcs need not be maximal arcs.