The Schläfli Fan
Michael Joswig, Marta Panizzut, and Bernd Sturmfels
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Submission date: 29. May. 2019
Link to arXiv: See the arXiv entry of this preprint.
Smooth tropical cubic surfaces are parametrized by maximal cones in the unimodular secondary fan of the triple tetrahedron. There are 344843867 such cones, organized into a database of 14373645 symmetry classes. The Schläfli fan gives a further refinement of these cones. It reveals all possible patterns of the 27 or more 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 algebraic geometry. We conclude with a sparse model for cubic surfaces over a field with valuation.