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\(\mathfrak{M}_{{\text{r,}}\varepsilon } \) 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 \(\overline {\mathcal{M}} _{0,n} \) 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 (Kahler) forms. This, via the link to \(\overline {\mathcal{M}} _{0,n} \), brings up a new tool to study the Kahler topology of\(\overline {\mathcal{M}} _{0,n} \). A wild but precise conjecture on the shape of the Kahler cone of \(\overline {\mathcal{M}} _{0,n} \) is given in the end.