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Building planar polygon spaces from the projective braid arrangement

Navnath Daundkar, Priyavrat Deshpande

Abstract

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 of the moduli space of distinct points on the projective line as an open dense subset. Hence, such a polygon space is a compactification of this real moduli space. Kapranov showed that the real points of the Deligne-Mumford-Knudson compactification can be obtained from the projective Coxeter complex of type $A$ (equivalently, the projective braid arrangement) by iteratively blowing up along the minimal building set. In this paper we show that these planar polygon spaces can also be obtained from the projective Coxeter complex of type $A$ by performing an iterative cellular surgery along a sub-collection of the minimal building set. Interestingly, this sub-collection is determined by the combinatorial data associated with the length vector called the genetic code.

Building planar polygon spaces from the projective braid arrangement

Abstract

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 of the moduli space of distinct points on the projective line as an open dense subset. Hence, such a polygon space is a compactification of this real moduli space. Kapranov showed that the real points of the Deligne-Mumford-Knudson compactification can be obtained from the projective Coxeter complex of type (equivalently, the projective braid arrangement) by iteratively blowing up along the minimal building set. In this paper we show that these planar polygon spaces can also be obtained from the projective Coxeter complex of type by performing an iterative cellular surgery along a sub-collection of the minimal building set. Interestingly, this sub-collection is determined by the combinatorial data associated with the length vector called the genetic code.
Paper Structure (10 sections, 25 theorems, 59 equations, 7 figures)

This paper contains 10 sections, 25 theorems, 59 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The braid arrangement, Coxeter complex and motivation
  3. The braid arrangement
  4. Motivation
  5. Planar polygon spaces
  6. Saturated chains
  7. Submanifold arrangements
  8. A theorem of Hausmann
  9. Combinatorial surgery on a meet semi-lattice
  10. Cellular surgery on simple cell complexes and the proof of Lg

Key Result

Theorem 1.1

Let $G$ be the genetic code of a length vector $\alpha=(\alpha_1,\dots,\alpha_m)$. Then the iterated cellular surgery on the Coxeter complex $CA_{m-2}$ (respectively, on projective Coxeter complex $\mathbb{P}CA_{m-2}$) along the elements of $\mathcal{G}_{\alpha}$ (respectively $\mathbb{P}\mathcal{G}

Figures (7)

  • Figure 2.1: The Coxeter complex $CA_{3}$ and the projective Coxeter complex $\mathbb{P}CA_{3}$
  • Figure 3.1: $\overline{\mathrm{K}}_{\langle5\rangle}\cong \mathbb{P}CA_{3}$ and $\mathcal{I}(\overline{\mathcal{A}_{\alpha}})\cong\Pi_{4}\setminus \{\hat{1}\}$
  • Figure 3.2: $\mathrm{K}_{\langle5\rangle}\cong CA_{3}$ with $\mathcal{I}(\mathrm{A}_{\alpha})$
  • Figure 5.1: Combinatorial surgery along $345$
  • Figure 6.1: Cellular surgery on $CA_{3}$ along $S_{123}$.
  • ...and 2 more figures

Theorems & Definitions (71)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.1
  • Definition 2.4
  • Definition 2.5
  • Example 2.2
  • Lemma 2.1
  • proof
  • ...and 61 more