Smooth tropical cubic surfaces from blow-up


Let $K$ be a valued field. A tropical cubic surface defined by a polynomial $f \in K[x,y,z]$ is smooth if the regular subdivision induced by the valuation of the coefficients on the Newton polytope of $f$ is a unimodular triangulation, i.e., the maximal cells in the subdivision are simplices of minimal volume $\frac{1}{6}$, see Chapter 4.5 of [1].

A smooth cubic surface in $\mathbb{P}^3$ is the blow up of $\mathbb{P}^2$ at six points. The rational map $\mathbb{P}^2 \dashrightarrow \mathbb{P}^3$ is induced by a basis of the space of ternary cubic forms through these six points.


Let $K$ be the field $\mathbb{Q}_2$ of $2$-adic numbers. The question is how to choose six points with integer coordinates and a basis for the space of ternary cubic forms through these six points such that the induced rational map gives a smooth tropical surface with respect to the 2-adic valuation.

Active problem!

This problem is currently being worked on by Marta Panizzut, Emre Sertöz and Bernd Sturmfels (Since mid-January 2019).

1. D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Graduate Studies in Mathematics, 161. American Mathematical Society, Providence, RI, 2015.
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