Maximal Arc - Properties

Properties

  • The number of lines through a fixed point p, not on a maximal arc K, intersecting K in d points, equals . Thus, d divides q.
  • In the special case of d = 2, maximal arcs are known as hyperovals which can only exist if q is even.
  • An arc K having one fewer point than a maximal arc can always be uniquely extended to a maximal arc by adding to K the point at which all the lines meeting K in d - 1 points meet.
  • In PG(2,q) with q odd, no non-trivial maximal arcs exist.
  • In PG(2,2h), maximal arcs for every degree 2t, 1 ≤ th exist.

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