Selected Open and Solved Problems in Computational Synthetic Geometry

Abstract

Computational Synthetic Geometry was the title of the Springer Lecture Notes of the first author with Bernd Sturmfels in 1989. During the last 25 years combinatorial structures such as abstract point-line configurations in the sense of Branko Grünbaum’s book from 2009 [17], $$(d-1)$$ -spheres of questionable convex d-polytopes, or regular maps have been studied in view of their possible geometric realization. We present selected open and solved problems from these areas. Oriented matroids have played an essential role in most of these problems. The topological representation of oriented matroids as sphere systems leads to pseudoline arrangements in the rank 3 case. We show in particular the application of a new topological representation of all combinatorial point-line configurations (quasiline arrangements).