Toric structure of the moduli space of points in projective space

TL;DR

This paper characterizes the fan structure of a toric compactification of the moduli space of points in projective space, revealing a symmetric product structure that generalizes classical combinatorial objects.

Contribution

It identifies the fan of the toric compactification as a symmetric product of a nested fan, extending the classical connection between Losev-Manin spaces and permutohedra to higher dimensions.

Findings

The fan of the toric compactification is a symmetric product of a nested fan.

Fans of Gallardo-Routis compactifications with reduction maps are symmetric products of building sets.

The combinatorics are governed by coarsenings of the permutohedral fan leading to Hassett spaces.

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.