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arXiv · 2508.06694

Tropical fans supporting a reduced 0-dimensional complete intersection

Linxuan Li
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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.

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Linxuan Li. 2025-08-08. Tropical fans supporting a reduced 0-dimensional complete intersection. https://arxiv.org/abs/2508.06694

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