Toric structure of the moduli space of points in projective space

TL;DR

The paper extends Hassett’s weighted moduli framework to configurations of points in $\mathbb{P}^d$ by analyzing Gallardo–Routis (GR) compactifications and their toric realizations. It identifies the toric fan of the Losev–Manin-type compactification $\overline{P}_{d,n}^{LM}$ as the $d$th symmetric product of a building-set-derived structure (a nested fan), and shows that any GR toric compactification admitting a reduction map from $\overline{P}_{d,n}^{LM}$ likewise corresponds to a symmetric product of a building set, thereby connecting to generalized permutohedra and Hassett-type coarsenings. A precise combinatorial description is provided via $P$-nestohedra and symmetric products $\operatorname{sym}^d(\mathcal{B})$, with LM corresponding to the complete building set on $\{d+2,\dots,n\}$; for admissible weight data, the GR compactifications are toric and reflect Hassett moduli spaces through a derived weight vector $\mathcal{A}'$. The results unify toric geometry, tropical geometry, and moduli theory by showing that the combinatorics governing these compactifications are controlled by coarsenings of the permutohedral fan, generalized to arbitrary dimension.

Abstract

Gallardo and Routis constructed compactifications of the moduli space of $n$ labeled points in $\mathbb{P}^d$ by assigning weights to points, generalizing Hassett's weighted compactifications of $M_{0,n}$ to higher-dimensional projective spaces. Among their compactifications, there is a toric compactification that generalizes the standard Losev-Manin compactification to this higher-dimensional setting. Our main result identifies the fan of this toric compactification as a symmetric product of a nested fan, generalizing the classical connection between Losev-Manin spaces and the permutohedron to arbitrary dimension. More generally, we prove that the fans of all Gallardo-Routis compactifications that admit reduction maps from this Losev-Manin space are symmetric products of building sets. This shows that the combinatorics of these compactifications are controlled by coarsenings of the permutohedral fan that give rise to Hassett spaces.

Figures (1)

• Figure 1: The fan $\operatorname{sym}^2(\mathcal{B})$ and corresponding polytope in Example \ref{['example sym prod']}.

Theorems & Definitions (31)

• Theorem 1.1: Theorem \ref{['main thm1']}
• Theorem 1.2: Theorem \ref{['main thm2']}
• Theorem 1.3: Theorem \ref{['main thm3']}
• Example 2.1
• Definition 2.2
• Proposition 2.3: gallardo2017wonderful
• Remark 2.4
• Definition 2.5
• Example 2.6
• Definition 2.7