View source for Arc (projective geometry)

[[File:Hyperoval in Fano plane.svg|thumb|The red points are a 4-arc in [[Fano plane]], the projective plane of order 2.]]
{{for|arcs in a real or complex projective space|Arc (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 geometry|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 {{math|'''''k'''''}}-'''arcs'''. An important generalization of {{mvar|k}}-arcs, also referred to as arcs in the literature, is the ({{mvar|k, d}})-arcs.

=={{mvar|k}}-arcs in a projective plane==

In a finite [[projective plane]] {{pi}} (not necessarily [[Desarguesian]]) a set {{mvar|A}} of {{math|''k'' (''k'' ≥ 3)}} points such that no three points of {{mvar|A}} are [[Collinear points|collinear]] (on a line) is called a {{math|'''''k'' - arc'''}}. If the plane {{pi}} has order {{mvar|q}} then {{math|''k'' ≤ ''q'' + 2}}, however the maximum value of {{mvar|k}} can only be achieved if {{mvar|q}} is even.<ref>{{harvnb|Hirschfeld|1979|loc=p. 164, Theorem 8.1.3}}</ref> In a plane of order {{mvar|q}}, a {{math|(''q'' + 1)}}-arc is called an '''[[Oval (projective plane)|oval]]''' and, if {{mvar|q}} is even, a {{math|(''q'' + 2)}}-arc is called a '''[[Oval (projective plane)|hyperoval]]'''.

Every conic in the Desarguesian projective plane PG(2,{{mvar|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 {{mvar|q}} is odd, every {{math|(''q'' + 1)}}-arc in PG(2,{{mvar|q}}) is a conic ([[Segre's theorem]]). This is one of the pioneering results in [[finite geometry]].

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

A {{mvar|k}}-arc which can not be extended to a larger arc is called a '''''complete arc'''''. In the Desarguesian projective planes, PG(2,{{mvar|q}}), no {{mvar|q}}-arc is complete, so they may all be extended to ovals.<ref>{{harvnb|Dembowski|1968|loc=p. 150, result 28}}</ref>

=={{mvar|k}}-arcs in a projective space==

In the finite [[projective space]] PG({{math|''n'', ''q''}}) with {{math|''n'' ≥ 3}}, a set {{mvar|A}} of {{math|''k'' ≥ ''n'' + 1}} points such that no {{math|''n'' + 1}}  points lie in a common [[Hyperplane (geometry)|hyperplane]] is called a (spatial) {{math|'''''k'''''}}-'''arc'''. This definition generalizes the definition of a {{mvar|k}}-arc in a plane (where {{math|1=''n'' = 2}}).

==({{math|''k'', ''d''}})-arcs in a projective plane==

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

The number of points {{mvar|k}} of a ({{math|''k'', ''d''}})-arc {{mvar|A}} in a projective plane of order {{mvar|q}} is at most {{math|''qd'' + ''d'' − ''q''}}.  When equality occurs, one calls {{mvar|A}} a '''[[maximal arc]]'''.

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

==See also==

* [[Normal rational curve]]

==Notes==
{{reflist}}

==References==
* {{Citation | last1=Dembowski | first1=Peter | title=Finite geometries | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Band 44 | mr=0233275 | year=1968 | isbn=3-540-61786-8 | url-access=registration | url=https://archive.org/details/finitegeometries0000demb }}
* {{citation|last=Hirschfeld|first=J.W.P.|title=Projective Geometries over Finite Fields|year=1979|publisher=Oxford University Press|location=New York|isbn=0-19-853526-0|url-access=registration|url=https://archive.org/details/projectivegeomet0000hirs}}

==External links==
*{{springer|id=Arc_(projective_geometry)&oldid=25358|title=Arc|author=C.M. O'Keefe}}

[[Category:Projective geometry]]
[[Category:Incidence geometry]]