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

TL;DR

This paper introduces stable polygons in Euclidean space, constructs their moduli spaces, and shows these are isomorphic to the moduli space of genus 0 stable curves, providing new symplectic tools for studying their topology.

Contribution

It constructs novel moduli spaces of stable polygons that are isomorphic to barM_{0,n} and introduces symplectic structures, offering new insights into the K"ahler topology of these moduli spaces.

Findings

Moduli spaces of stable polygons are projective and isomorphic to barM_{0,n}.

Stable polygons naturally induce symplectic (K"ahler) forms.

These spaces serve as symplectic counterparts to known moduli spaces, aiding topological study.

Abstract

In this paper, certain natural and elementary polygonal objects in Euclidean space, {\it the stable polygons}, are introduced, and the novel moduli spaces ${\bfmit M}_{{\bf r}, \epsilon}$ 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 $\bar{\cM}_{0,n}$ of the Deligne-Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data leading toward to a family of (classes of) symplectic (K\"ahler) forms. To some degree, ${\bfmit M}_{{\bf r}, \epsilon}$ may be considered as symplectic counterparts of $\bar{\cM}_{0,n}$ and Kapranov's Chow quotient construction of $\bar{\cM}_{0,n}$. All these together brings up a new tool to study the K\"ahler topology of $\bar{\cM}_{0,n}$.