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Effective faithful tropicalizations associated to linear systems on curves

About this Title

Shu Kawaguchi and Kazuhiko Yamaki

Publication: Memoirs of the American Mathematical Society
Publication Year: 2021; Volume 270, Number 1323
ISBNs: 978-1-4704-4753-3 (print); 978-1-4704-6534-6 (online)
DOI: https://doi.org/10.1090/memo/1323
Published electronically: June 28, 2021
Keywords: Algebraic curves, linear system, faithful tropicalization, skeleton, Berkovich space, nonarchimedean geometry

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Table of Contents

Chapters

• 1. Introduction
• 2. Preliminaries
• 3. Good models
• 4. Unimodular tropicalization of minimal skeleta for $g \geq 2$
• 5. Faithful tropicalization of minimal skeleta for $g \geq 2$
• 6. Faithful tropicalization of minimal skeleta in low genera
• 7. Faithful tropicalization of arbitrary skeleta
• 8. Complementary results
• 9. Limit of tropicalizations by polynomials of a bounded degree

Abstract

For a connected smooth projective curve $X$ of genus $g$, global sections of any line bundle $L$ with $\deg (L) \geq 2g+ 1$ give an embedding of the curve into projective space. We consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry: We replace projective space by tropical projective space, and an embedding by a homeomorphism onto its image preserving integral structures (or equivalently, since $X$ is a curve, an isometry), which is called a faithful tropicalization.

Let $K$ be an algebraically closed field which is complete with respect to a non-trivial nonarchimedean value. Suppose that $X$ is defined over $K$ and has genus $g \geq 2$ and that $\Gamma$ is a skeleton (that is allowed to have ends) of the analytification $X^{\mathrm {an}}$ of $X$ in the sense of Berkovich. We show that if $\deg (L) \geq 3g-1$, then global sections of $L$ give a faithful tropicalization of $\Gamma$ into tropical projective space.

As an application, when $Y$ is a suitable affine curve, we describe the analytification $Y^{\mathrm {an}}$ as the limit of tropicalizations of an effectively bounded degree.