Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The $S_n$-equivariant Euler characteristic of $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$

Siddarth Kannan, Terry Dekun Song

Abstract

We compute the $S_n$-equivariant topological Euler characteristic of the Kontsevich moduli space $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$. Letting $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d) \subset \overline{\mathcal{M}}_{1, n}(¶^r, d)$ denote the subspace of maps from curves without rational tails, we solve for the motive of $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$ in terms of $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d)$ and plethysm with a genus-zero contribution determined by Getzler and Pandharipande. Fixing a generic $\mathbb{C}^\star$-action on $\mathbb{P}^r$, we derive a closed formula for the Euler characteristic of $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d)^{\mathbb{C}^\star}$ as an $S_n$-equivariant virtual mixed Hodge structure, which leads to our main formula for the Euler characteristic of $\overline{\mathcal{M}}_{1,n}(\mathbb{P}^r, d)$. Our approach connects the geometry of torus actions on Kontsevich moduli spaces with symmetric functions in Coxeter types $A$ and $B$, as well as the enumeration of graph colourings with prescribed symmetry.

The $S_n$-equivariant Euler characteristic of $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$

Abstract

We compute the -equivariant topological Euler characteristic of the Kontsevich moduli space . Letting denote the subspace of maps from curves without rational tails, we solve for the motive of in terms of and plethysm with a genus-zero contribution determined by Getzler and Pandharipande. Fixing a generic -action on , we derive a closed formula for the Euler characteristic of as an -equivariant virtual mixed Hodge structure, which leads to our main formula for the Euler characteristic of . Our approach connects the geometry of torus actions on Kontsevich moduli spaces with symmetric functions in Coxeter types and , as well as the enumeration of graph colourings with prescribed symmetry.

Paper Structure

This paper contains 32 sections, 30 theorems, 202 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Graded $\mathbb{S}$-spaces and plethysm
  3. $\mathbb{C}^\star$-localisation and type-$B$ symmetric functions
  4. Enumerative combinatorics of cycles
  5. History and context
  6. Further directions
  7. Symmetric functions and wreath products
  8. Symmetric functions and $S_n$-equivariant characteristics
  9. $\mathbb{S}$-modules
  10. $S_2\times S_n$-symmetric functions
  11. Type-$B$ symmetric functions
  12. $S_2 \times \mathbb{S}$ and $\mathbb{B}$-modules
  13. Plethysm in the graded setting
  14. Composition and localisation on graded $\mathbb{S}$-spaces
  15. $\mathbb{S}$-, $\mathbb{B}$-, and $S_2 \times \mathbb{S}$-spaces
  16. ...and 17 more sections

Key Result

Theorem A

For $r \geq 1$, we have where The factors $\eta_{k,d}(r)$ and $\theta_{j, k,d}(r)$ are polynomials in $\mathbb{Q}[r]$ given by the formulas and

Figures (4)

  • Figure 1: The marking assignment is in black, and the degree assignment in red. Vertices without red labels have degree zero.
  • Figure 2: A six-cycle with vertex and half-edge decorations. We use $i$ in place of $v_i$ in the figure.
  • Figure 3: A labeled decorated graph in $\tilde{\Gamma}_{r, 6}(6)^{\langle \rho^2, \rho\tau\rangle}$ with degree decoration in red and vertex colouring is given by red and black dots. Its quotient graph under $\langle \rho^2, \rho\tau\rangle$-symmetry is on the right.
  • Figure 4: Two distinct labeled colourings with symmetry group $\langle \rho^2\rangle$. Their quotient graphs become the same after forgetting the edge orderings.

Theorems & Definitions (55)

  • Theorem A
  • Theorem : Proposition \ref{['prop-cut']}
  • Theorem : Theorem \ref{['thm-locnrt']}
  • Theorem B
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 45 more