Higher-dimensional Losev-Manin spaces and their geometry

TL;DR

This paper explores higher-dimensional generalizations of Losev-Manin spaces, revealing their geometric structure, fiber bundle nature, and conditions under which they are Mori dream spaces, with implications for moduli space theory.

Contribution

It introduces higher-dimensional toric moduli spaces, describes their fibration structure, and provides criteria for when these spaces are Mori dream spaces, including a counterexample for certain moduli spaces.

Findings

Higher-dimensional Losev-Manin spaces form a fibration over a product of projective spaces.

These spaces are isomorphic to the normalization of a Chow quotient.

A criterion is given for when blow-ups of toric varieties are Mori dream spaces.

Abstract

The classical Losev-Manin space is a toric compactification of the moduli space of $n$ points in the affine line modulo translation and scaling. Motivated by this, we study its higher-dimensional toric counterparts, which compactify the moduli space of $n$ distinct labeled points in affine space modulo translation and scaling. We show that these moduli spaces are a fibration over a product of projective spaces -- with fibers isomorphic to the Losev-Manin space -- and that they are isomorphic to the normalization of a Chow quotient. Moreover, we present a criterion to decide whether the blow-up of a toric variety along the closure of a subtorus is a Mori dream space. As an application, we demonstrate that a related generalization of the moduli space of pointed rational curves constructed by Chen, Gibney, and Krashen is not a Mori dream space when the number of points is at least nine, regardless of the dimension.