LEADER 00821nam0-22003011i-450- 001 990007684800403321 005 20141002112121.0 035 $a000768480 035 $aFED01000768480 035 $a(Aleph)000768480FED01 035 $a000768480 100 $a20030814d--------km-y0itay50------ba 101 0 $aita 200 1 $a<>EEC and Intellectual Property$fDiana Guy$gGuy I. F. Leigh 210 $aLondon$cSweet & Maxwell$d1981 215 $aXXIV,375 p.$d24 cm 676 $a340.2 700 1$aGuy,$bDiana$0231160 702 1$aLeigh,$bGuy I. F. 801 0$aIT$bUNINA$gRICA$2UNIMARC 901 $aBK 912 $a990007684800403321 952 $aDI VIII 352$b7000$fDEC 952 $a3-F-10$b4371$fDDCP 959 $aDEC 959 $aDDCP 996 $aEEC and Intellectual Property$9680648 997 $aUNINA LEADER 05438nam 22006253 450 001 9910970819103321 005 20231110230425.0 010 $a9781470465346 010 $a1470465345 035 $a(CKB)4100000011975798 035 $a(MiAaPQ)EBC6661102 035 $a(Au-PeEL)EBL6661102 035 $a(OCoLC)1259591042 035 $a(RPAM)22488291 035 $a(EXLCZ)994100000011975798 100 $a20210901d2021 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aEffective Faithful Tropicalizations Associated to Linear Systems on Curves 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2021. 210 4$dİ2021. 215 $a1 online resource (122 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.270 311 08$a9781470447533 311 08$a1470447533 320 $aIncludes bibliographical references and index. 327 $aCover -- 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 ?-metric graphs -- 3.3. Weighted ?-metric graphs -- 3.4. Skeleton as a weighted ?-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]. 327 $aChapter 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. 330 $a"For a connected smooth projective curve of genus g, global sections of any line bundle L with deg(L) 2g 1 give an embedding of the curve into projective space. We consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry: We replace projective space by tropical projective space, and an embedding by a homeomorphism onto its image preserving integral structures (or equivalently, since is a curve, an isometry), which is called a faithful tropicalization. Let be an algebraically closed field which is complete with respect to a nontrivial nonarchimedean value. Suppose that is defined over and has genus g 2 and that is a skeleton (that is allowed to have ends) of the analytification an of in the sense of Berkovich. We show that if deg(L) 3g 1, then global sections of L give a faithful tropicalization of into tropical projective space. As an application, when Y is a suitable affine curve, we describe the analytification Y an as the limit of tropicalizations of an effectively bounded degree"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aGeometry, Algebraic 606 $aTropical geometry 606 $aAlgebraic geometry -- Tropical geometry -- Tropical geometry$2msc 606 $aAlgebraic geometry -- Arithmetic problems. Diophantine geometry -- Rigid analytic geometry$2msc 606 $aAlgebraic geometry -- Cycles and subschemes -- Divisors, linear systems, invertible sheaves$2msc 615 0$aGeometry, Algebraic. 615 0$aTropical geometry. 615 7$aAlgebraic geometry -- Tropical geometry -- Tropical geometry. 615 7$aAlgebraic geometry -- Arithmetic problems. Diophantine geometry -- Rigid analytic geometry. 615 7$aAlgebraic geometry -- Cycles and subschemes -- Divisors, linear systems, invertible sheaves. 676 $a516.3/52 686 $a14T05$a14G22$a14C20$2msc 700 $aKawaguchi$b Shu$01799882 701 $aYamaki$b Kazuhiko$01799883 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910970819103321 996 $aEffective Faithful Tropicalizations Associated to Linear Systems on Curves$94344303 997 $aUNINA