Weight functions on Berkovich curves

Baker, Matthew; Nicaise, Johannes

doi:10.2140/ant.2016.10.2053

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Weight functions on Berkovich curves

Matthew Baker and Johannes Nicaise

Vol. 10 (2016), No. 10, 2053–2079

DOI: 10.2140/ant.2016.10.2053

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

Vol. 10, No. 10, 2016