Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement “any smooth surface of degree 3 in P$^3$ contains exactly 27 lines” is known to be false tropically. Work of Vigeland from 2007 provides examples of cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP$^3$.

In this talk I will explain how to correct this pathology. The novel idea is to consider the embedding of a smooth cubic surface in P$^4$$^4$ via its anticanonical bundle. The tropicalization induced by this embedding contains exactly 27 lines under a mild genericity assumption. More precisely, smooth cubic surfaces in P$^3$ are del Pezzos, and can be obtained by blowing up P$^2$ at six points in general position. We identify these points with six parameters over a field with nontrivial valuation. Our genericity assumption involves the valuations of 36 linear expressions in these parameters which give the positive roots of type E$_6$. Tropical convexity plays a central role in ruling out the existence of extra tropical lines on the tropical cubic surface.

This talk is based on an ongoing project joint with Anand Deopurkar.