Tropical Lines on Cubic Surfaces

2007/08/28 by Panizzut, Marta, Vigeland, Magnus Dehli
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.0708.3847

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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