  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.
