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Automorphisms of the boundary complex of $\overline{\mathcal{M}}_{0, n}(\mathbb{P}^r, d)$

Arjun Joisha, Siddarth Kannan

Abstract

We compute the automorphism group of the dual complex $\mathsf{T}_{d, n}$ of the boundary divisor in the Kontsevich moduli space $\overline{\mathcal{M}}_{0, n}(\mathbb{P}^r, d)$. When $d \geq 2$, we find that $\mathrm{Aut}(\mathsf{T}_{d, n}) \cong \mathbb{S}_{n}$, while $\mathrm{Aut}(\mathsf{T}_{1, n}) \cong \mathbb{S}_{n + 1}$ for all $n \geq 4$. The complex $\mathsf{T}_{1, n}$ is also the dual complex of the boundary divisor in the Fulton--MacPherson compactification of the configuration space of $n$ points on $X$, if $X$ is any smooth, proper, and connected algebraic variety over $\mathbb{C}$. Following work of Massarenti, this implies that $\mathsf{T}_{1, n}$ admits automorphisms which in general do not extend to $X[n]$.

Automorphisms of the boundary complex of $\overline{\mathcal{M}}_{0, n}(\mathbb{P}^r, d)$

Abstract

We compute the automorphism group of the dual complex of the boundary divisor in the Kontsevich moduli space . When , we find that , while for all . The complex is also the dual complex of the boundary divisor in the Fulton--MacPherson compactification of the configuration space of points on , if is any smooth, proper, and connected algebraic variety over . Following work of Massarenti, this implies that admits automorphisms which in general do not extend to .

Paper Structure

This paper contains 14 sections, 18 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. The boundary complex
  4. A strategy for computing $\operatorname{Aut}(\mathsf{T}_{d, n})$
  5. The contraction deck
  6. Preservation of vertex weights
  7. A reconstruction algorithm
  8. Node types
  9. Partitioning edges and markings
  10. Case I
  11. Case II
  12. Trees with two vertices
  13. The case $d = 1$ and the Fulton--MacPherson compactification
  14. From abstract simplicial complexes to symmetric $\Delta$-complexes

Key Result

Theorem 1.1

When $d \geq 2$, we have for all $n$. When $d = 1$, we have for $n \geq 4$, while $\operatorname{Aut}(\mathsf{T}_{1, 3}) \cong \mathbb{S}_3$ and $\operatorname{Aut}(\mathsf{T}_{1, n})$ is trivial for $n \leq 2$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Distinct $2$-cells of $\mathsf{T}_{12, 0}$ which have the same ordered list of $1$-dimensional faces, corresponding to distinct codimension-$3$ strata of the normalization of the boundary of $\overline{\mathcal{M}}_{0, 0}(\mathbb{P}^r, 12)$ which are not determined by their local branches. The vertex-weighted trees represent dual graphs of stable maps of degree $12$: each vertex corresponds to an irreducible component of the source curve, and the integer weights record the degree of the map on the corresponding component.
  • Figure :
  • Figure :

Theorems & Definitions (40)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Lemma 3.1
  • Lemma 3.2
  • ...and 30 more