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arXiv · 0708.3847

Tropical Lines on Cubic Surfaces

Marta Panizzut·Magnus Dehli Vigeland
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Abstract

Given a tropical line $L$ and a smooth tropical surface $X$, we look at the position of $L$ on $X$. We introduce its primal and dual motif which are respectively a decorated graph and a subcomplex of the dual triangulation of $X$. They encode the combinatorial position of $L$ on $X$. We classify all possible motifs of tropical lines on general smooth tropical surfaces. This classification allows to give an upper bound for the number of tropical lines on a general smooth tropical surface with a given subdivision. We focus in particular on surfaces of degree three. As a concrete example, we look at tropical cubic surfaces dual to a fixed honeycomb triangulation, showing that a general surface contains exactly $27$ tropical lines.

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Marta Panizzut, Magnus Dehli Vigeland. 2019-04-16. Tropical Lines on Cubic Surfaces. https://arxiv.org/abs/0708.3847

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