Arc (projective geometry)
Script error: No such module "For". 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 Template:Math-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 Template:Math points such that no three points of Template:Mvar are collinear (on a line) is called a Template:Math. If the plane Template:Pi has order Template:Mvar then Template:Math, however the maximum value of Template:Mvar can only be achieved if Template:Mvar is even.[1] In a plane of order Template:Mvar, a Template:Math-arc is called an oval and, if Template:Mvar is even, a Template:Math-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 Template:Math-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 Template:Math-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.[2]
Template:Mvar-arcs in a projective space
In the finite projective space PG(Template:Math) with Template:Math, a set Template:Mvar of Template:Math points such that no Template:Math points lie in a common hyperplane is called a (spatial) Template:Math-arc. This definition generalizes the definition of a Template:Mvar-arc in a plane (where Template:Math).
(Template:Math)-arcs in a projective plane
A (Template:Math)-arc (Template:Math) 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 (Template:Math)-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 (Template:Math)-arc Template:Mvar in a projective plane of order Template:Mvar is at most Template:Math. When equality occurs, one calls Template:Mvar a maximal arc.
Hyperovals are maximal arcs. Complete arcs need not be maximal arcs.
See also
Notes
References
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