Isoresidual fibrations and the birational geometry of moduli of pointed stable curves

TL;DR

The paper addresses the birational geometry of the moduli space $\overline{M}_{0,n}$ by proving the pseudo-effective cone is not polyhedral for $n\ge 8$, thereby showing $\overline{M}_{0,n}$ is not a Mori Dream Space in this range. The authors introduce residue maps on strata of meromorphic differentials and resolve their extensions via multi-scale compactifications $\Xi\overline{\mathcal{M}}_{0,n}(\mu)$, enabling a Stein factorisation analysis and the construction of a non-polyhedral extremal moving-ray $F_{\mu}$. They identify resonance transform divisors $D_S^{\mu}$ that are rigid and extremal, and compute their classes, establishing a rich provable structure of the effective cone tied to resonance geometry. The results have significant implications for understanding the birational models of $\overline{M}_{0,n}$ and demonstrate limitations on the Mori Dream Space property beyond small $n$, with broader connections to flat geometry, Gauss-Manin theory, and moduli of differentials. The work also foregrounds multi-scale techniques as a powerful tool for resolving maps that interact with boundary combinatorics.

Abstract

We show that the pseudo-effective cone of divisors of $\overline{M}_{0,n}$ is not polyhedral for $n\geq 8$ by constructing an extremal non-polyhedral ray of the dual cone of moving curves via maps on meromorphic strata of differentials returning the residues at the poles of the parameterised differentials. An immediate corollary is that these spaces are not Mori Dream Spaces.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.4
• Theorem 1.5
• Theorem 2.1: Theorem 1.7 CGPT24
• Definition 3.1
• Definition 3.2
• Theorem 3.3