Arc (projective geometry)

File:Hyperoval in Fano plane.svg

The red points are a 4-arc in Fano plane, the projective plane of order 2.

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 Template:Mvar-arcs, also referred to as arcs in the literature, is the (Template:Mvar)-arcs.

Template:Mvar-arcs in a projective plane

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

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

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

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

Template:Mvar-arcs in a projective space

In the finite projective space PG(n, q) with n ≥ 3, a set Template:Mvar 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 Template:Mvar-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 Template:Pi (not necessarily Desarguesian) is a set, Template:Mvar of Template:Mvar points of Template:Pi such that each line intersects Template:Mvar in at most Template:Mvar points, and there is at least one line that does intersect Template:Mvar in Template:Mvar points. A (k, 2)-arc is a Template:Mvar-arc and may be referred to as simply an arc if the size is not a concern.

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

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

See also

• Normal rational curve

External links

• Template:Springer