Small resolutions of moduli spaces of scaled curves

TL;DR

The paper constructs small resolutions of the singular moduli spaces ${\overline{Q}_n}$ and ${\overline{P}_n}$ of stable scaled curves and $\mathbb{G}_a$-rational trees by embedding them into smooth ambient spaces $W_n$ and $W'_n$, which are augmented wonderful varieties built from polydiagonal arrangements in ${\mathbb P}^{n-1}$. Central to the construction are polydiagonal degenerations of $X^n$ and $X[n]$, realized via Li’s wonderful compactifications, together with a root relative dualizing sheaf $\sqrt[d]{\omega}_{W,X/S}$ that provides a scaling on fibers. The main results establish the existence of small resolutions $\gamma: W_n \to \overline{P}_n$ and $\eta: W'_n \to \overline{Q}_n$ for $n\ge 4$ (with isomorphisms for $n\le 3$), thereby producing smooth, SNC-bounded compactifications that reflect the combinatorics of polydiagonal degenerations and connect to matroid-driven augmented wonderful varieties. The work links these resolutions to Fulton–MacPherson-type degeneration spaces and opens avenues for generalizations to other matroids and deeper analyses of their fiber structures, while providing concrete geometric models that refine known resolutions of matroid Schubert varieties. Overall, the constructions yield explicit small-resolutions of key moduli spaces, with a robust interplay between combinatorics, degeneration theory, and a root-scaled geometric framework.

Abstract

We construct small resolutions of the moduli space $\overline{Q}_n$ of stable scaled $n$-marked lines of Ziltener and Ma'u--Woodward and of the moduli space $\overline{P}_n$ of stable $n$-marked ${\mathbb G}_a$-rational trees introduced in earlier work. The resolution of $\overline{P}_n$ is the augmented wonderful variety corresponding to the graphic matroid of the complete graph. The resolution of $\overline{Q}_n$ is a further blowup, also a wonderful model of an arrangement in ${\mathbb P}^{n-1}$.

Figures (3)

• Figure 1: The two types of boundary divisors of $\overline{Q}_n$. The picture on the left also corresponds to a boundary divisor of $\overline{P}_n$.
• Figure 2: For $n=4$: the arrangement ${\mathcal{P}}oly_H$ if $H$ is the screen or paper (left); the lines and point in ${\mathcal{D}}iag$ (right).
• Figure 3: ${\mathcal{H}}= \{123|45678,12|3|45|67|8,12|3|45|6|7|8\}$. We are not contracting all the way to the leg (visible for legs $3,6,7,8$), in contrast to [Ul02]. Instead, we contract only up to the corresponding leaf, even when there is a single leg attached to it.

Theorems & Definitions (120)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Example 2.7
• Remark 2.8