Arc (projective geometry)

An (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 the k-arc concept, also referred to as arcs in the literature, are the (k, d)-arcs.

A 4-arc (red points) in the projective plane of order 2 (Fano plane).
For arcs in a real or complex projective space, see Arc (geometry).

k-arcs in a projective plane

In a finite projective plane π (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.[1] 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.[2]

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.

See also

Notes
  1. Hirschfeld 1979, p. 164, Theorem 8.1.3
  2. Dembowski 1968, p. 150, result 28

References
  • Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275
  • Hirschfeld, J.W.P. (1979), Projective Geometries over Finite Fields, New York: Oxford University Press, ISBN 0-19-853526-0
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