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
Milestones
Received: 28 October 2015
Revised: 24 June 2016
Accepted: 2 October 2016
Published: 9 December 2016
Authors
Matthew Baker
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
United States
Johannes Nicaise
Department of Mathematics
KU Leuven
Celestijnenlaan 200B
3001 Heverlee
Belgium
Department of Mathematics
Imperial College
South Kensington Campus
London
SW7 2AZ
United Kingdom