Valuatively independent bases for the Fermat family of cubic curves

Let $\pi :(X,L)\to {\mathbb{D}}^{\ast }$ be the Fermat family of cubic curves in ${\mathbb{P}}^2$. For each $k\ge 1$, we construct a valuatively independent basis for $H^0(X,L^k)$. The construction uses a canonical cost function determined by a Hessian structure on the essential skeleton $\text{op}Sk(X,\pi )$.