Tropical fans supporting a reduced 0-dimensional complete intersection

TL;DR

This paper classifies regular affine tropical fans supporting reduced 0-dimensional complete intersections, introduces the concept of galleries for classification, and proves finiteness results in dimension 2.

Contribution

It introduces the notion of galleries to classify one-dimensional regular fans and establishes finiteness theorems for certain two-dimensional fans.

Findings

Classification of all one-dimensional regular fans using galleries.

Finiteness of regular fans in dimension 2 under specific conditions.

Minimal model program for one-dimensional regular fans.

Abstract

An affine tropical fan is called regular if it supports a reduced 0-dimensional complete intersection. For some cases the classification of regular fans is already complete. It was proved by Fink that tropical varieties of degree 1 are exactly Bergman fans, and later Esterov and Gusev classified all lattice polytopes whose mixed volume equals 1. We introduce the notion of a gallery for tropical fans and use it to classify all one-dimensional regular fans, thereby obtaining a minimal model programme for such fans. In dimension 2 we prove a finiteness theorem: every regular fan that satisfies the given upper bound condition is precisely the support of a finite covering by two-dimensional galleries, and only finitely many such fans exist.