Realization spaces for tropical fans

TL;DR

This paper develops a framework for understanding the moduli of algebraic varieties with prescribed tropicalizations, showing that these spaces are algebraic and exploring their geometric properties, especially for matroid fans.

Contribution

It introduces a moduli functor for tropical realizations, proves it is an algebraic space, and analyzes the structure of realization spaces for matroid fans.

Findings

Tropical realization spaces are algebraic spaces and schemes of finite type in certain cases.

The realization space of a matroid fan is a torus torsor over the matroid's realization space.

These spaces satisfy Murphy's Law, indicating complex and unpredictable geometric behavior.

Abstract

We introduce a moduli functor for varieties whose tropicalization realizes a given weighted fan and show that this functor is an algebraic space in general, and is represented by a scheme of finite type when the associated toric variety is quasiprojective. We study the geometry of these tropical realization spaces for the matroid fans studied by Ardila and Klivans, and show that the tropical realization space of a matroid fan is a torus torsor over the realization space of the matroid. As a consequence, we deduce that these tropical realization spaces satisfy Murphy's Law.