Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The dual complex of $\mathcal{M}_{1,n}(\mathbb{P}^r,d)$ via the geometry of the Vakil--Zinger moduli space

Siddarth Kannan, Terry Dekun Song

Abstract

We study normal crossings compactifications of the moduli space of maps $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$, for $g = 0$ and $g = 1$. In each case we explicitly determine the dual boundary complex, and prove that it admits a natural interpretation as a moduli space of decorated metric graphs. We prove that the dual complexes are contractible when $r \geq 1$ and $d > g$. When $g = 1$, our result depends on a new understanding of the connected components of boundary strata in the Vakil--Zinger desingularization and its modular interpretation by Ranganathan--Santos-Parker--Wise.

The dual complex of $\mathcal{M}_{1,n}(\mathbb{P}^r,d)$ via the geometry of the Vakil--Zinger moduli space

Abstract

We study normal crossings compactifications of the moduli space of maps , for and . In each case we explicitly determine the dual boundary complex, and prove that it admits a natural interpretation as a moduli space of decorated metric graphs. We prove that the dual complexes are contractible when and . When , our result depends on a new understanding of the connected components of boundary strata in the Vakil--Zinger desingularization and its modular interpretation by Ranganathan--Santos-Parker--Wise.

Paper Structure

This paper contains 18 sections, 22 theorems, 101 equations, 9 figures.

Table of Contents

  1. Combinatorial preliminaries
  2. Radial alignments on genus $1$ graphs
  3. Modular compactifications of mapping spaces
  4. radially-aligned prestable curves
  5. radially-aligned stable maps
  6. Geometry of graph strata
  7. Graph strata of $\mathfrak{M}_{1, n}^\mathrm{rad}$
  8. Strata of mapping spaces
  9. Irreducibility of strata
  10. Parameterized maps with a fixed tangent vector
  11. Strata of the Kontsevich moduli space
  12. Strata as fiber products
  13. Topology of dual complexes
  14. The virtual dual complex $\Delta_{g, n}^{\mathrm{vir}}(d)$
  15. Contractibility of the virtual dual complex
  16. ...and 3 more sections

Key Result

Theorem A

Fix $r \geq 1$. The stratum $\widetilde{\mathcal{M}}(\mathbf{G}, \rho)$ is irreducible, and there is an explicit combinatorial criterion for when $\widetilde{\mathcal{M}}(\mathbf{G}, \rho)$ is nonempty. The dual complex of the divisor is naturally identified with the symmetric $\Delta$-complex $\Delta_{1, n}(d)$.

Figures (9)

  • Figure 1: An example of a stable $(2, 2, 9)$-graph $\mathbf{G}$. Red numbers next to vertices indicate the value of $\delta$ on the vertex; other vertices are assumed to have $\delta$-value equal to $0$. The vertex with a purple $1$ is assumed to have $w$-value $1$, while other vertices have $w = 0$. Finally, the function $m$ is visualized by adding labeled half-edges.
  • Figure 2: A stable radially-aligned $5$-marked graph $\mathbf{G}$ of genus $1$, degree $7$, and length $3$. All vertices have $g(v) = 0$, and the red labels indicate the $\delta$ degree of a vertex. The blue boxes indicate the fibers of the surjection $V(\mathbf{G}) \to \{0, 1, 2, 3\}$, and the labeled half-edges indicate the marking function $m$.
  • Figure 3: The canonical subdivision $\hat{\mathbf{G}}$ of the graph $\mathbf{G}$ from Figure \ref{['fig:comb-type-exmp']}, along with the map $\hat{\rho}: \hat{\mathbf{G}} \to P_{3}$ induced by the radial alignment.
  • Figure 4: The subdivision $\hat{\mathbf{G}}_1$ at radius $1$ of the graph $\mathbf{G}$ from Figure \ref{['fig:comb-type-exmp']}, along with the induced radial alignment.
  • Figure 5: From left to right: the radial merges of the graph $\mathbf{G}$ from Figure \ref{['fig:comb-type-exmp']} along $1$, $2$, and $3$, above the corresponding canonical subdivisions and maps to $P_2$.
  • ...and 4 more figures

Theorems & Definitions (76)

  • Theorem A
  • Theorem B
  • Corollary C
  • Remark
  • Theorem : Theorem \ref{['thm:virtual-contraction']}
  • Proposition : Proposition \ref{['prop:subspace_preserved']}
  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • ...and 66 more