# Markov triples and Newton-Okounkov bodies of $\mathbb{P}^2$

I am working on symplectic geometry and I have some questions about a degeneration of $$\mathbb{P}^2$$.

Question: Can we obtain the moment polytope (or the polytope associated with the anti-canonical divisor) of $$\mathbb{P}(a^2,b^2,c^2)$$ as a Newton-Okounkov body of $$\mathbb{P}^2$$? (Here $$(a,b,c)$$ is a Markov triple satisfying $$a^2 + b^2 + c^2 = 3abc$$.)

The motivation of this question is as follows: It is known by Hacking-Prokhorov that the weighted projective space $$\mathbb{P}(a^2,b^2,c^2)$$ admits a $$\mathbb{Q}$$-Gorenstein smoothing with a generic fiber $$\mathbb{P}^2$$ where $$(a,b,c)$$ is a Markov triple. I want to understand a $$\mathbb{Q}$$-Gorenstein smoothing of $$\mathbb{P}(a^2,b^2,c^2)$$ with generic fiber $$\mathbb{P}^2$$ as a toric degeneration of $$\mathbb{P}^2$$ with the central fiber $$\mathbb{P}(a^2,b^2,c^2)$$. Dave Anderson proved that if a polytope $$P$$ can be realized as a Newton-Okounkov body with a certain condition (finitely generatedness of a semigroup), then there exists a toric degeneration of $$X$$ with the central fiber $$X_0$$ where $$X_0$$ is a toric variety whose normalization is a normal toric variety associated with the polytope $$P$$.

I would really appreciate for any comment. Thank you!

The polytopes you are interested in are related by sequences of combinatorial mutations, as described here and here. If two polytopes $$P_1$$ and $$P_2$$ are related by a combinatorial mutation, then there is a construction due to Ilten (here) of a flat family $$\pi\colon\mathcal X\to \mathbb P^1$$ such that $$\pi^{-1}(0)=X_{P_1}$$ is the toric variety defined by the spanning fan of $$P_1$$ and $$\pi^{-1}(\infty)=X_{P_2}$$ is the toric variety defined by the spanning fan of $$P_2$$. You can interpret the toric degeneration of $$\mathbb P^2$$ to $$\mathbb P (a^2,b^2,c^2)$$ as a following a sequence of 1-parameter families by moving along the relevant edges of the Markov tree.