Weight functions on Berkovich curves

TL;DR

This paper explores the properties of weight functions on Berkovich curves, providing tropical descriptions and linking them to the essential skeleton, with implications for understanding curve reductions.

Contribution

It introduces tropical descriptions of weight functions and the essential skeleton on Berkovich curves, connecting them to pluricanonical forms and minimal models.

Findings

Laplacian of the weight function equals the pluricanonical divisor

Essential skeleton coincides with the minimal skeleton under semi-stable reduction

Describes base loci of pluricanonical line bundles on minimal models

Abstract

Let $C$ be a curve over a complete discretely valued field $K$. We give tropical descriptions of the weight function attached to a pluricanonical form on $C$ and the essential skeleton of $C$. We show that the Laplacian of the weight function equals the pluricanonical divisor on Berkovich skeleta, and we describe the essential skeleton of $C$ as a combinatorial skeleton of the Berkovich skeleton of the minimal $snc$-model. In particular, if $C$ has semi-stable reduction, then the essential skeleton coincides with the minimal skeleton. As an intermediate step, we describe the base loci of logarithmic pluricanonical line bundles on minimal $snc$-models.