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| [[File:Hyperoval in Fano plane.svg|thumb|A 4-arc (red points) in the projective plane of order 2 (Fano plane).]]
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| 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 the ''k''-arc concept, also referred to as arcs in the literature, are the (''k'', ''d'')-arcs.
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| ==''k''-arcs in a projective plane==
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| In a finite [[projective plane]] ''π'' (not necessarily [[Desarguesian]]) a set ''A'' of ''k'' (''k'' ≥ 3) points such that no three points of ''A'' are [[Collinear points|collinear]] (on a line) is called a '''''k'''''-'''arc'''. If the plane ''π'' has order ''q'' then ''k'' ≤ ''q'' + 2, however the maximum value of ''k'' can only be achieved if ''q'' is even.<ref>{{harvnb|Hirschfeld|1979|loc=pg. 164, Theorem 8.1.3}}</ref> In a plane of order ''q'', a (''q'' + 1)-arc is called an '''[[Oval (projective plane)|oval]]''' and, if ''q'' is even, a (''q'' + 2)-arc is called a '''[[Oval (projective plane)|hyperoval]]'''.
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| 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.<ref>{{harvnb|Dembowski|1968|loc=pg. 150, result 28}}</ref>
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| ==''k''-arcs in a projective space==
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| In the finite [[projective space]] PG(''n'', ''q'') with ''n'' ≥ 3, a set ''A'' of ''k'' ≥ ''n'' + 1 points such that no ''n'' + 1 points lie in a common [[Hyperplane (geometry)|hyperplane]] is called a (spacial) '''''k'''''-'''arc'''. This definition generalizes the definition of a ''k''-arc in a plane (where ''n'' = 2).
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| ==(''k'', ''d'')-arcs in a projective plane==
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| A (''k'', ''d'')-'''arc''' (''k'', ''d'' > 1) in a finite [[projective plane]] ''π'' (not necessarily [[Desarguesian]]) is a set, ''A'' of ''k'' points of <math>\pi</math> 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.
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| 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]]'''.
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| Hyperovals are maximal arcs. Complete arcs need not be maximal arcs.
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| ==See also==
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| [[Oval (projective plane)]] | |
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| [[Normal rational curve]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{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}}
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| * {{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}}
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| ==External links==
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| *{{springer|id=Arc_(projective_geometry)&oldid=25358|title=Arc|author=C.M. O'Keefe}}
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| [[Category:Projective geometry]]
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