This repository contains the code belonging to the master thesis "Computing Lines on Tropical Cubic Surfaces" written by Lena Weis and supervised by Prof. Dr. Michael Joswig (TU Berlin) and Dr. Marta Panizzut (MPI Mathematics in Sciences). It was published in order to make the results of the thesis FAIR, i.e. findable, accessible, interoperable and reusable.
The aim of this thesis was to compute the number of lines on a surface of one of five different combintorial types of smooth tropical cubic surface utilizing the Schläfli fan introduced in [1]. They are
- #37
- #842554
- #2091566
- #5054117
- #12369387
The Schläfli fans can be found here and the larger datasets are stored here.
First include the secondary cone and Motifs of the triangulation of interest. They can be found here. Include the algorithm to compute the Schläfli walls are print them with this.
Include this file and procede as in the following example
Which triangulation?
2091566
What are the coefficients of f defining the surface? (Write as x0 x1 x2...)
19 -4 -13 -12 2 -7 -11 3 -10 12 -5 -28 -27 -6 -10 8 8 -10 7 21
The corresponding surface has this many visible motifs of motifs 3A, 3B, 3C, 3D, 3E, 3H, i.e. lines:
21
Note that you can only check this for one of the five triangulations above. For triangulation #842554 we have included one cone of the Schläfli fan such that the number of lines on the surfaces dual to the points in each cone is exactly 14, 15, 17, 18, 19, 20 or 21.
The remaining statements for each triangulation in the thesis, including the number of lines, are computed [here](Computations/Code/Computating\ lines).
- Note to 5054117: We further checked whether the polynomial (2) in [2] is actually dual to the triangulation 5054117.
[1] Joswig, Michael and Panizzut, Marta and Sturmfels, Bernd. (2020). The Schläfli Fan. A Discrete Comput Geom 64, pp. 355–381
[2] Hampe, Simon and Joswig, Michael. (2018). Tropical Computations in polymake. Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, pp. 361–385.