Vol. 40, No. 3, 1972

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Simple points in pseudoline arrangements

Leroy Milton Kelly and R. Rottenberg

Vol. 40 (1972), No. 3, 617–622

A finite collection of simple closed curves in the real projective plane each two of which have exactly one point in common at which point they “cross” is called a pseudoline arrangement. Such arrangements seem to have been first systematically studied by Levi. Recently B. Grünbaum has called attention to the desirability for a better understanding of the differences as well as the similarities in the behavior of arrangements of lines and the arrangements of pseudolines. Among other things, Grünbaum asks if every pseudoline arrangement, not all curves of which intersect in a single point, must have a simple vertex (a point on exactly two of the curves of the arrangement) as is the case in line arrangements. In fact Kelly and Moser have shown that, in general, an n-line arrangement in an ordered projective plane has at least 37n simple vertices. It is shown here that their reasoning can be nearly dualized to prove the analogous result for pseudoline arrangements.

Mathematical Subject Classification
Primary: 50D20
Received: 16 February 1971
Revised: 16 June 1971
Published: 1 March 1972
Leroy Milton Kelly
R. Rottenberg