Effective Faithful Tropicalizations Associated to Linear Systems on Curves

• Autore: Kawaguchi Shu
• Pubblicazione: Providence : , : American Mathematical Society, , 2021; ©2021
• Descrizione fisica: 1 online resource (122 pages)
• Disciplina: 516.3/52
• Soggetto topico: Geometry, Algebraic; Tropical geometry; Algebraic geometry -- Tropical geometry -- Tropical geometry; Algebraic geometry -- Arithmetic problems. Diophantine geometry -- Rigid analytic geometry; Algebraic geometry -- Cycles and subschemes -- Divisors, linear systems, invertible sheaves
• Classificazione: 14T0514G2214C20
• Altri autori: YamakiKazuhiko
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: 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.
• Sommario/riassunto: "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"--
• Titolo autorizzato: Effective Faithful Tropicalizations Associated to Linear Systems on Curves
• ISBN: 1-4704-6534-5
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910915736803321
• Lo trovi qui: Univ. Federico II
• Serie: Memoirs of the American Mathematical Society