Valuatively independent bases for the Fermat family of cubic curves

TL;DR

This paper constructs valuatively independent bases for sections of line bundles on Fermat cubic curves using a canonical cost function derived from a Hessian structure on the skeleton.

Contribution

It introduces a novel construction of valuatively independent bases for all tensor powers of the line bundle on Fermat cubic curves, utilizing Hessian structures.

Findings

Constructed bases are valuatively independent for all k ≥ 1.

Uses a canonical cost function from Hessian structures on the skeleton.

Provides a new approach to understanding sections of line bundles on 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 $\op{Sk}(X,\pi)$.