Irrational toric varieties and secondary polytopes

TL;DR

This paper extends the theory of toric varieties to irrational settings, linking degenerations and secondary polytopes, and develops a new framework for irrational toric varieties associated with arbitrary fans.

Contribution

It introduces a theory of irrational toric varieties for arbitrary fans, generalizing classical toric varieties and connecting them to secondary polytopes.

Findings

Irrational toric varieties are homeomorphic to polytopes when associated with normal fans.

The space of degenerations of irrational toric varieties corresponds to secondary polytopes.

The theory unifies rational and irrational cases within a common framework.

Abstract

The space of torus translations and degenerations of a projective toric variety forms a toric variety associated to the secondary fan of the integer points in the polytope corresponding to the toric variety. This is used to identify a moduli space of real degenerations with the secondary polytope. A configuration A of real vectors gives an irrational projective toric variety in a simplex. We identify a space of translations and degenerations of the irrational projective toric variety with the secondary polytope of A. For this, we develop a theory of irrational toric varieties associated to arbitrary fans. When the fan is rational, the irrational toric variety is the nonnegative part of the corresponding classical toric variety. When the fan is the normal fan of a polytope, the irrational toric variety is homeomorphic to that polytope.