# Khovanskii bases of Cox-Nagata rings and tropical geometry

## TL;DR

This paper explores the structure of Cox rings of degree 3 del Pezzo surfaces, focusing on Khovanskii bases and their relation to tropical geometry, extending previous classifications from degree 4 cases.

## Contribution

It introduces a classification problem for degree 3 Cox rings, analyzes tropical geometric structures involved, and demonstrates limitations of certain tropical fans for this classification.

## Key findings

- Initial forms of Cox ring generators do not always generate the initial algebra.
- Tropical Grassmannian $ ext{TGr}(3, extbf{Q}^6)$ and Naruki fan are insufficient for classification.
- The paper connects Cox ring structures with tropical geometry concepts.

## Abstract

The Cox ring of a del Pezzo surface of degree 3 has a distinguished set of 27 minimal generators. We investigate conditions under which the initial forms of these generators generate the initial algebra of this Cox ring. Sturmfels and Xu provide a classification in the case of degree 4 del Pezzo surfaces by subdividing the tropical Grassmannian $\operatorname{TGr}(2,\mathbb{Q}^5)$. After providing the necessary background on Cox-Nagata rings and Khovanskii bases, we review the classification obtained by Sturmfels and Xu. Then we describe our classification problem in the degree 3 case and its connections to tropical geometry. In particular, we show that two natural candidates, $\operatorname{TGr}(3,\mathbb{Q}^6)$ and the Naruki fan, are insufficient to carry out the classification.