Arc (projective geometry)

A 4-arc (red points) in the projective plane of order 2 (Fano plane).

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.

$k$ -arcs in a projective plane

In a finite projective plane $π$ (not necessarily Desarguesian) a set $A$ of $k (k \geq 3)$ 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 $k \leq q + 2$ , 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 $n \geq 3$ , a set $A$ of $k \geq n + 1$ 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.

Notes

↑ Hirschfeld 1979, p. 164, Theorem 8.1.3
↑ 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

External links

C.M. O'Keefe (2001), "Arc", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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Arc (projective geometry)

k-arcs in a projective plane

k-arcs in a projective space

(k, d)-arcs in a projective plane

See also