Tropical fans supporting a reduced 0-dimensional complete intersection

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.

💡 Research Summary

The paper studies affine tropical fans that are “regular”, meaning they support a reduced 0‑dimensional complete intersection (i.e., there exist k non‑negative tropical regular functions whose tropical intersection product with the fan yields the unit 0‑cycle). This notion generalizes smooth tropical fans (Bergman fans) and the classification of lattice polytopes of mixed volume 1. The author introduces two new concepts—galleries and a minimal model programme (MMP)—to systematically describe regular fans.

First, regular cycles are defined: a k‑dimensional affine tropical cycle