Towards classifying toric degenerations of cubic surfaces

Maria Donten-Bury, Paul Görlach, and Milena Wrobel

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Submission date: 26. Sep. 2019
Keywords and phrases: cubic surface, del Pezzo surface, Cox ring, Toric degeneration, tropicalization
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
We investigate degenerations of cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. 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 leave the question of classifying these degenerations in the degree 3 case as an open problem. We describe an approach for the classification which is closely related to tropical geometry and present partial results in this direction.