The Space of Unordered Real Line Arrangements

The set of all unordered real line arrangements of given degree in the real projective plane is known to have a natural semialgebraic structure. The nonreduced arrangements are singular points of this structure. We show that the set of all unordered real line arrangements of given degree also has a natural structure of a smooth compact connected affine real algebraic variety. In fact, as such, it is isomorphic to a real projective space. As a consequence, we get a projectively linear structure on the set of all real line arrangements of given degree. We also show that the universal family of unordered real line arrangements of given degree is not algebraic.