Valuatively independent bases for the Fermat family of cubic curves

Abstract

Let $\pi:(X,L)\rightarrow \mathbb D^*$ be the Fermat family of cubic curves in $\mathbb P^2$. For each $k\geq 1$, 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 $\op{Sk}(X,\pi)$.