Effective Faithful Tropicalizations Associated to Linear Systems on Curves.

Format:

Book

Author/Creator:

Kawaguchi, Shu.

Contributor:

Yamaki, Kazuhiko.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.270

Language:

English

Subjects (All):

Geometry, Algebraic.

Tropical geometry.

Physical Description:

1 online resource (122 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

"For a connected smooth projective curve of genus g, global sections of any line bundle L with deg(L) 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 is a curve, an isometry), which is called a faithful tropicalization. Let be an algebraically closed field which is complete with respect to a nontrivial nonarchimedean value. Suppose that is defined over and has genus g 2 and that is a skeleton (that is allowed to have ends) of the analytification an of in the sense of Berkovich. We show that if deg(L) 3g 1, then global sections of L give a faithful tropicalization of into tropical projective space. As an application, when Y is a suitable affine curve, we describe the analytification Y an as the limit of tropicalizations of an effectively bounded degree"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

Notation and Conventions

Chapter 2. Preliminaries

2.1. Semistable models and semistable pairs

2.2. Berkovich spaces

2.3. Skeleta associated to strictly semistable models

2.4. Skeleta associated to strictly semistable pairs

2.5. Some properties of skeleta

2.6. Tropical geometry

2.7. Faithful tropicalization

Chapter 3. Good models

3.1. Good models of

3.2. Theory of divisors on Λ-metric graphs

3.3. Weighted Λ-metric graphs

3.4. Skeleton as a weighted Λ-metric graph (with a finite graph structure)

3.5. Construction of a model of ( , )

Chapter 4. Unimodular tropicalization of minimal skeleta for ≥2

4.1. Useful lemmas

4.2. Fundamental vertical divisors

4.3. Stepwise vertical divisors

4.4. Edge-base sections and edge-unimodularity sections

4.5. Unimodular tropicalization

Chapter 5. Faithful tropicalization of minimal skeleta for ≥2

Notation and terminology of Chapter 5

5.1. Separating points on an edge of connected type

5.2. Separating points in different edges

5.3. Separating vertices

5.4. Faithful tropicalization of the minimal skeleton

Chapter 6. Faithful tropicalization of minimal skeleta in low genera

6.1. Genus 0 case

6.2. Genus 1 case

Chapter 7. Faithful tropicalization of arbitrary skeleta

Notation and terminology of Chapter 7

7.1. Geodesic paths

7.2. Stepwise vertical divisor associated to a point in ( )

7.3. Base sections and -unimodularity sections

7.4. Good model

7.5. Proof of Proposition 7.8

7.6. Proof of Theorem 1.2

7.7. Upper bound for the dimension of the target space

Chapter 8. Complementary results

8.1. Theorem 1.2 is optimal for curves in low genera

8.2. A very ample line bundle that does not admit a faithful tropicalization

8.3. Comparison with [42].

Chapter 9. Limit of tropicalizations by polynomials of a bounded degree

9.1. Statement of the result

9.2. Polynomial of bounded degree that separates two points

9.3. Proof of Theorem 1.7

Bibliography

Subject Index

Symbol Index

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references and index.

ISBN:

9781470465346

1470465345

OCLC:

1259591042