  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.
