Real Tropical Singularities and Bergman Fans

TL;DR

This paper classifies singular real plane tropical curves using subdivisions of Newton polytopes, introduces signed Bergman fans, and explores their combinatorial and parametric properties, advancing understanding of real tropical geometry.

Contribution

It introduces signed Bergman fans and a duality framework for real tropical curves, providing a classification of singular curves of maximal dimensional type.

Findings

Signed Bergman fans generalize positive Bergman fans.

Duality between real tropical curves and signed subdivisions established.

Classification of singular real plane tropical curves of maximal dimensional type.

Abstract

In this paper, we classify singular real plane tropical curves by means of subdivisions of Newton polytopes. First, we introduce signed Bergman fans (generalizing positive Bergman fans from [AKW06]) that describe real tropicalizations of real linear spaces ([Tab15]). Then, we establish a duality of real plane tropical curves and signed regular subdivisions of the Newton polytope and explore the combinatorics. We define a signed secondary fan that parametrizes real tropical Laurent polynomials and study the subset providing singular real plane tropical curves. A cone of the signed secondary fan is of maximal dimensional type if its corresponding subdivision contains only marked points ([MMS12a]). These cones parametrize real plane tropical curves. We classify singular real plane tropical curves of maximal dimensional type.