Arc (projective geometry)

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 (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.{{harvnb|Hirschfeld|1979|loc=p. 164, Theorem 8.1.3}} In a plane of order {{mvar|q}}, a {{math|(q + 1)}}-arc is called an oval and, if {{mvar|q}} is even, a {{math|(q + 2)}}-arc is called a 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.{{harvnb|Dembowski|1968|loc=p. 150, result 28}}

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 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}}).

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.

• Normal rational curve

{{reflist}}

• {{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}}

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

Category:Projective geometry

Category:Incidence geometry