Towards classifying toric degenerations of cubic surfaces

TL;DR

This paper explores the classification of toric degenerations of smooth cubic surfaces using Khovanskii bases and tropical geometry, extending previous work on Del Pezzo surfaces of degree 4.

Contribution

It introduces an approach combining Khovanskii bases and tropical geometry to classify toric degenerations of cubic surfaces, addressing an open problem.

Findings

Partial classification results obtained.

Connection established between Khovanskii bases and tropical geometry.

Framework proposed for future complete classification.

Abstract

We investigate the class of degenerations of smooth cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. More precisely, we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and who leave the question of classifying these degenerations in the degree 3 case as an open problem. In order to carry out this classification we describe an approach which is closely related to tropical geometry and present partial results in this direction.