Higher-dimensional Losev-Manin spaces and their geometry

Patricio Gallardo, Javier González-Anaya, José Luis González, Evangelos Routis
Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG)
2023-08-14 16:00:00
The classical Losev-Manin space can be interpreted as 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 proposed by Chen, Gibney, and Krashen is not a Mori dream space when the number of points is at least nine, regardless of the dimension.
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