05438nam 22006253 450 991097081910332120231110230425.097814704653461470465345(CKB)4100000011975798(MiAaPQ)EBC6661102(Au-PeEL)EBL6661102(OCoLC)1259591042(RPAM)22488291(EXLCZ)99410000001197579820210901d2021 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierEffective Faithful Tropicalizations Associated to Linear Systems on Curves1st ed.Providence :American Mathematical Society,2021.©2021.1 online resource (122 pages)Memoirs of the American Mathematical Society ;v.2709781470447533 1470447533 Includes bibliographical references and index.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."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.Memoirs of the American Mathematical Society Geometry, AlgebraicTropical geometryAlgebraic geometry -- Tropical geometry -- Tropical geometrymscAlgebraic geometry -- Arithmetic problems. Diophantine geometry -- Rigid analytic geometrymscAlgebraic geometry -- Cycles and subschemes -- Divisors, linear systems, invertible sheavesmscGeometry, 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.516.3/5214T0514G2214C20mscKawaguchi Shu1799882Yamaki Kazuhiko1799883MiAaPQMiAaPQMiAaPQBOOK9910970819103321Effective Faithful Tropicalizations Associated to Linear Systems on Curves4344303UNINA