Moduli Spaces of Stable Polygons and Symplectic Structures on$\overline{\mathcal M}$0,n

Abstract

In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces${mathfrak M}$$_r,ϵ$ of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space $\overlinel{\matcal M}$$_0,n$ of the Deligne–Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data giving rise to a family of (classes of) symplectic (Kähler) forms. This, via the link to $\overlinel{\matcal M}$$_0,n$, brings up a new tool to study the Kähler topology of$\overlinel{\matcal M}$$_0,n$. A wild but precise conjecture on the shape of the Kähler cone of $\overlinel{\matcal M}$$_0,n$is given in the end.