Anticanonical tropical cubic del Pezzos contain exactly 27 lines

TL;DR

This paper proves that tropical cubic del Pezzo surfaces, under generic conditions, contain exactly 27 tropical lines, paralleling classical geometry, and explores their moduli space and combinatorial structure.

Contribution

It establishes the exact number of tropical lines on generic tropical cubic del Pezzo surfaces and describes their moduli space using tropical geometry and root system combinatorics.

Findings

Generic tropical cubic del Pezzo surfaces contain exactly 27 tropical lines.

Non-generic cases have up to 27 extra lines, which do not lift to algebraic curves.

The moduli space is a four-dimensional fan with Weyl group symmetry.

Abstract

The classical statement of Cayley-Salmon that there are 27 lines on every smooth cubic surface in P^3 fails to hold under tropicalization: a tropical cubic surface in TP^3 often contains infinitely many tropical lines. Under mild genericity assumptions, we show that when embedded using the Eckardt triangles in the anticanonical system, tropical cubic del Pezzo surfaces contain exactly 27 tropical lines. In the non-generic case, which we identify explicitly, we find up to 27 extra lines, no multiple of which lifts to a curve on the cubic surface. We realize the moduli space of stable anticanonical tropical cubics as a four-dimensional fan in R^40 with an action of the Weyl group W(E_6). In the absence of Eckardt points, we show the combinatorial types of these tropical surfaces are determined by the boundary arrangement of 27 metric trees corresponding to the tropicalization of the classical 27 lines on the smooth algebraic cubic surfaces. Tropical convexity and the combinatorics of the root system E_6 play a central role in our analysis.