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

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

1997-01-27

arxiv.org/abs/dg-ga/9701011v2

Mathematics

Algebraic Geometry

Enlarged version, 7 figures added. 35 pages. To appear in Compositio Mathematica

Scientific paper

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 $\bar{\cM}_{0,n}$.

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