Small resolutions of moduli spaces of scaled curves

TL;DR

This paper constructs small resolutions of certain moduli spaces of scaled curves, providing new geometric models that clarify their structure and relate to wonderful varieties and arrangements.

Contribution

It introduces explicit small resolutions for the moduli spaces ar{Q}_n and ar{P}_n, connecting them to wonderful varieties and graphic matroids.

Findings

Resolved ar{P}_n as an augmented wonderful variety.

Resolved ar{Q}_n via a further blowup, relating to arrangements in projective space.

Established geometric models linking moduli spaces to combinatorial structures.

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 ${\mathbb P}^{n-1}$.