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The moduli space of planar polygons with generic side lengths is a smooth,closed manifold. It is known that these manifolds contain the real points ofthe moduli space of distinct points on the projective line as an open densesubset. Hence, such a polygon space is a compactification of this real modulispace. Kapranov showed that the real points of the Deligne-Mumford-Knudsoncompactification can be obtained from the projective Coxeter complex of type$A$ (equivalently, the projective braid arrangement) by iteratively blowing upalong the minimal building set. In this paper we show that these planar polygonspaces can also be obtained from the projective Coxeter complex of type $A$ byperforming an iterative cellular surgery along a sub-collection of the minimalbuilding set. Interestingly, this sub-collection is determined by thecombinatorial data associated with the length vector called the genetic code.

Building planar polygon spaces from the projective braid arrangement