Tropical fans supporting a reduced 0-dimensional complete intersection
Combinatorics
2025-12-19 v2 Algebraic Geometry
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.
Cite
@article{arxiv.2508.06694,
title = {Tropical fans supporting a reduced 0-dimensional complete intersection},
author = {Linxuan Li},
journal= {arXiv preprint arXiv:2508.06694},
year = {2025}
}