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

Question

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!

Answer 1

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.