Pascalian Configurations in Projective Planes

Abstract

Publisher Summary If Ω is a projective oval of a projective plane π and if P (Ω) be the figure formed by all of the secants or tangents to omega, which are pascalian lines with respect to omega, then the figures P (Ω) are called “Ω-pascalian configurations” of π. The problem of determining the configuration P (Ω) have been introduced, and by using these configurations, it is possible to produce an interesting classification for the projective ovals. If P (Ω) is a Ω-pascalian configuration of one of the four known projective plane of order 4: Desargues, Hughes, and two Hall planes, then P (Ω) coincides with one of the specific sets outlined in the chapter. Such sets are defined as all the tangents and all the secants, the empty set, and all the tangents and all the secants through a unique point. There are projective ovals having non-exterior pascalian lines in non-desarguesian planes, but in this setting, the problem is very far from being resolved. At present, two other Ω-pascalian configurations have been found.