UNIVERSITY OF DELAWARE
DEPARTMENT OF MATHEMATICAL SCIENCES
DISCRETE MATHEMATICS SEMINAR

Monday, June 1, 3 pm, Room 436 Ewing Hall

Ovals in finite projective planes

Gábor Korchmáros, Universita della Basilicata (Italy)

The classical concept of a conic leads in a natural way to the concept of an oval in an arbitrary projective plane: An oval is a subset tex2html_wrap_inline11 of points satisfying both of the following properties: i) no three points of tex2html_wrap_inline11 are collinear; ii) tex2html_wrap_inline11 has exactly one 1-secant (also called a tangent) at each one of its points. If the plane is finite and has order n, then an oval consists of n+1 points. Ovals of finite projective planes have been intensively studied since 1954. The starting point was the famous theorem of B. Segre: In a Desarguesian plane of odd order, the ovals are exactly the irreducible conics. This talk is a survey of known results in the following topics:
  1. The classification problem for ovals in a desarguesian plane of even order.
  2. Ovals in finite non desarguesian planes.
  3. Pascal's theorem for ovals and abstract ovals.
  4. Collineation groups fixing an oval; some characterizations of the finite desarguesian planes.