Maria Angelica Cueto: Anticanonical tropical del Pezzo surfaces contain exactly 27 lines

Abstract: 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 three 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^44 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 is joint work in progress with Anand Deopurkar.