# The Schl\"afli Fan

## TL;DR

The paper introduces the Schl"afli fan, a detailed combinatorial structure that refines the classification of tropical cubic surfaces, revealing all line patterns and serving as a foundational tool for tropical geometry data analysis.

## Contribution

It develops the theory of the Schl"afli fan and provides a framework for analyzing large-scale data in tropical geometry.

## Key findings

- Refined classification of tropical cubic surfaces via the Schl"afli fan
- Identification of all possible line patterns on tropical cubic surfaces
- A blueprint for big data analysis in tropical geometry

## Abstract

Smooth tropical cubic surfaces are parametrized by maximal cones in the unimodular secondary fan of the triple tetrahedron. There are $344\, 843 \,867$ such cones, organized into a database of $14\,373\,645$ symmetry classes. The Schl\"afli fan gives a further refinement of these cones. It reveals all possible patterns of lines on tropical cubic surfaces, thus serving as a combinatorial base space for the universal Fano variety. This article develops the relevant theory and offers a blueprint for the analysis of big data in tropical geometry.