#### Vol. 10, No. 10, 2016

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Weight functions on Berkovich curves

### Matthew Baker and Johannes Nicaise

Vol. 10 (2016), No. 10, 2053–2079
##### 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 semistable 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.

##### Keywords
Berkovich spaces, degenerations of curves, tropical geometry
##### Mathematical Subject Classification 2010
Primary: 14D10
Secondary: 14E30, 14T05