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The Gauss--skizze decomposition is a Goresky-MacPherson stratification

N. C. Combe

Abstract

We consider a new stratification of the space of configurations of $n$ marked points on the complex plane. Recall that this space can be differently interpreted as the space $^{\rm D}{\rm Pol}_{n}$ of degree $n>1$ complex, monic polynomials with distinct roots, the sum of which is 0. A stratum $A_σ$ is the set of polynomials having $P^{-1}(\mathbb{R}\cup\imath\mathbb{R})$ in the same isotopy class, relative to their asymptotic directions. We show that this stratification is a Goresky--MacPherson stratification and that from thickening strata a good cover in the sense of Čech can be constructed, allowing an explicit computation of the cohomology groups of this space.

The Gauss--skizze decomposition is a Goresky-MacPherson stratification

Abstract

We consider a new stratification of the space of configurations of marked points on the complex plane. Recall that this space can be differently interpreted as the space of degree complex, monic polynomials with distinct roots, the sum of which is 0. A stratum is the set of polynomials having in the same isotopy class, relative to their asymptotic directions. We show that this stratification is a Goresky--MacPherson stratification and that from thickening strata a good cover in the sense of Čech can be constructed, allowing an explicit computation of the cohomology groups of this space.

Paper Structure

This paper contains 26 sections, 31 theorems, 21 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A new point of view on $\overline{\mathcal{M}}_{0,T}$
  3. Moduli spaces of genus 0 curves with marked points
  4. Definition-construction
  5. We consider the set of $n+1$ marked points on $\mathbb{P}^1$ as a configuration space of $n$ points on the complex plane modulo the group $PSL_2(\mathbb{C})$. This configuration space $Conf_{n}(\mathbb{C})$ can itself be considered as the set of monic complex polynomials, with $n$ roots, which we denote $\mathop{{}^{\textsc{D}}\text{Pol}}_{n}$. The quotient of this space by $PSL_2(\mathbb{C})$ allows to map three of the marked points to $\{0,1,\infty\}$.
  6. For the construction of this new stratification, it is very important to have a fixed real structure on $\mathbb{P}^1$. The main idea of our construction is to take the inverse image, under a polynomial $P$ in $\mathop{{}^{\textsc{D}}\text{Pol}}_n$, of the real and imaginary axis, i.e. $P^{-1}(\mathbb{R}\cup\imath \mathbb{R})$. This inverse image forms a system of oriented curves in the complex plane. To distinguish $P^{-1}(\mathbb{R})$ from $P^{-1}(\mathbb{R}\imath\mathbb{R})$, we color in red the pre-image of $\mathbb{R}$ and in blue the pre-image of $\mathbb{R}\imath\mathbb{R}$. The orientation of the curves is inherited from the natural orientation on the real and imaginary axis (see Figure \ref{['F:patdisc']}).
  7. Embedded forests
  8. From real curves to forests
  9. From real curves to stable trees
  10. Stratification on $\overline{\mathcal{M}}_{0,n}$
  11. Real stratification of the parametrizing space
  12. Some properties of the stratification
  13. $\mathop{{}^{\textsc{D}}\text{Pol}}_{d}$ as a covering of a non-compact stratified space
  14. Critical values
  15. The cases $n=2,3,4$
  16. ...and 11 more sections

Key Result

Theorem 1.0.1

Consider the configuration space of $n$ marked points on the complex plane, where points are pairwise disjoint. Then, there exists a real algebraic stratification $\mathcal{S}$ of this space, where strata are indexed by oriented and bicolored forests verifying the following properties:

Figures (9)

  • Figure 1: Partition of the complex plane
  • Figure 2: $P(z)= (z+1.54*(0.98+i*0.2))*(z+0.68*(0.98+i*0.2))*(z+0.4*(0.98+i*0.2))*(z-1.54*(0.98+i*0.2))*(z-0.68*(0.98+i*0.2))*(z-0.4*(0.98+i*0.2)).$
  • Figure 3: Desingularizing a double point
  • Figure 4: Example of a complete contracting Whitehead move
  • Figure 5: Example of two partial contracting Whitehead moves (done simultaneously)
  • ...and 4 more figures

Theorems & Definitions (73)

  • Definition 1.0.1: Čech cover Cech32
  • Theorem 1.0.1
  • Theorem 1.0.2
  • Theorem 1.0.3
  • Definition 2.2.1
  • Remark 1
  • Definition 2.2.2
  • Definition 2.2.3
  • Definition 2.2.4: Codimension
  • Example 1
  • ...and 63 more