Tropical fans supporting a reduced 0-dimensional complete intersection
Linxuan Li
TL;DR
The paper develops a framework for regular tropical cycles, achieving a full classification of regular 1-cycles via a universal projection to a direct sum of Bergman fans and a minimal model programme. It introduces galleries to organize regular sequences and provides an explicit description of all nonnegative tropical regular functions with intersection 1. A finiteness theorem for strongly regular 2-cycles under Hodge-type bounds is proved, with examples illustrating the sharpness of hypotheses and connections to engineered complete intersections. The results offer structural certificates and a constructive pathway for understanding irreducibility phenomena in toric and tropical complete intersections.
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.