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

TL;DR

This paper investigates the structure of boundary complexes of moduli spaces of maps from genus 0 and 1 curves to projective space, revealing their contractibility and a modular interpretation via decorated graphs.

Contribution

It explicitly determines the dual boundary complexes for these moduli spaces and interprets them as moduli spaces of decorated metric graphs, advancing understanding of their topology.

Findings

Dual complexes are contractible for r ≥ 1 and d > g.

Boundary complexes can be interpreted as moduli spaces of decorated graphs.

New insights into the Vakil--Zinger desingularization and its modular structure.

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.