Families of pointed toric varieties and degenerations

TL;DR

This paper explores a generalization of the Losev-Manin moduli space to families of pointed degenerate toric varieties, analyzing their geometric properties through combinatorial polytope fibrations.

Contribution

It introduces a new class of polytope fibrations generalizing twisted Cayley sums to describe families of pointed degenerate toric varieties.

Findings

Families are characterized by specific polytope fibrations.

Geometric properties like flatness and reduced fibers are established.

Connections to permutohedra and extremal contractions are demonstrated.

Abstract

The Losev-Manin moduli space parametrizes pointed chains of projective lines. In this paper we study a possible generalization to families of pointed degenerate toric varieties. Geometric properties of these families, such as flatness and reducedness of the fibers, are explored via a combinatorial characterization. We show that such families are described by a specific type of polytope fibration which generalizes the twisted Cayley sums, originally introduced to characterize elementary extremal contractions of fiber type associated to projective $\mathbb{Q}$-factorial toric varieties with positive dual defect. The case of a one-dimensional simplex can be viewed as an alternative construction of the permutohedra.