Algebraic geometry II PDF
Preview Algebraic geometry II
TEXTSAND READINGS 73 IN MATHEMATICS Algebraic Geometry 11 Texts and Readin_s in Mathematics Advisory Editor C. S.Seshadri, Chennai Mathematical Institute, Chennai. ManagingEditor Rajendra Bhatia, Indian Statistical Institute, New Delhi. Editors Manindra Agrawal,Indian Institute of Technology,Kanpur. V.Balaji, ChennaiMathematical Institute, Chennai. R. B. Bapat,Indian Statistical Institute, New Delhi. V.S.Borkar,Indian Institute of Technology,Mumbai. T. R. Ramadas,Chennai Mathematical Institute, Chennai. V...Srinivas, Tata Inst. of Fundamental Research,Mumbai. TechnicalEditor P.Vanchinathan, Vellore Institute of Technology,Chennai. Algieraic Geometry 11 David Mumford Brown University, USA Tadao .da Tohoku University, Japan HINIUSTAN BOOKAGENCY Publishedby HindustanBookAgency(India) P 19Green Park Extension New Delhi 110016 India email:[email protected] www.hindbook.com COpyrigh©t 2015,HindustanBookAgency(India) Nopartofthematerialprotectedbythiscopyrightnoticemaybe reproducedorutilizedin anyformorbyanymeans,electronicor mechanical,including.photocopying,recording by information or any storageandretrievalsystemw, ithoutwrittenpermissionfromthe copyrightowner,whohasalsothesoleright tograntlicencesfor translationintootherlanguageasndpublicationthereof. A11exportrightsforthiseditionvestexclusivelywith Hindustan BookAgency(India).Unauthorizedexportisaviolation of CopyrightLawandissubjecttolegalaction. ISBN 978-93-80250-80-9 Contents Preface ix 1 Schemesand sheaves:dennitions 1 1.1 Spec(R) 1 .............................. fl? 1.2 8 ................................. - 1.3 Schemes 11 .............................. 1.4 Products 18 ............................. 1.5 Quasi-coherent sheaves 24 ..................... 1.6 The functor of points 32 ...................... 1.7 Relativization 37 .......................... 1.8 Denning schemesas functors 40 .................. Appendix: Theory of sheaves 47 ...................... Exercises 52 ................................. 2 Exploring the world of schemes 57 2.1 Classical varieties asschemes 57 .................. 2.2 The properties: reduced, irreducible and nnite type 63 ...... 2.3 Closedsubschemesand primary decompositions 72 ....... 2.4 Separated schemes 81 ........................ 2.5 Proj R 86 ............................... 2.6 Pr0per morphisms 95 ........................ Exercises 103 ................................. 3 Elementary global study of Pro j R 105 3.1 Invertible sheavesand twists 105 .................. 3.2 The functor 0f Proj R 111 ...................... 3.3 Blowups 118 .............................. 3.4 Quasi-coherentsheavesoniProj R 122 ............... 3.5 Ample invertible sheaves 129 .................... 3.6 Invertible sheavesvia.cocycles, divisors, line bundles 136 ..... Exercises 142 ................................. vi Contents 4 Ground neldsand baserings 151 4.1 Kronecker’s big picture 151 ..................... 4.2 Galois theory and schemes 157 ................... 4.3 The Frobenjus morphism 171 .................... 4.4 Flatness and specialization 175 ................... 4.5 Dimension of nbresof morphism 184 a ............... 4.6 Hensel’s lemma 192 .......................... Exercises 197 ................................. 5 Singular vs. non-singular 199 5.1 Regularity 199 ............................ 5.2 Kéhler differential 204 ........................ 5.3 Smooth morphisms 212 ....................... 5.4 Criteria for smoothness 222 ..................... 5.5 Normality 233 ............................. 5.6 Zariski’s Main Theorem 239 ..................... 5.7 Multiplicities following Weil 249 ................... Exercises 253 ................................. 6 Group schemes and applications 261 6.1 Group schemes 261 .......................... 6.2 Lang’s theorems over nnite nelds 279 ................ Exercises 284 ................................. 7 The cohomology of coherent sheaves 287 7.1 Basic Cech cohomology 287 ..................... 7.2 The case of schemes: Serre’s theorem 299 .............. 7.3 Higher direct images and Leray’s spectral sequence 307 ...... 7.4 Computing cohomology (1): Push .7:into a hugeacyclic sheaf 316 7.5 Computing cohomology(2): Directly via the Cechcomplex 319 . 7.6 Computing cohomology (3): GenerateJ: by “known” sheaves 324 7.7 Computing cohomology(4): Push33into a coherentacyclic sheaf 328 ............................... 7.8 Serre’s criterion for ampleness 336 . . ................ 7.9 Functorial properties of ampleness 338 ............... 7.10 The Euler characteristic 342 ..................... 7.11 Intersection numbers 345 ....................... 7.12 The criterion of Nakai-Moishezon 348 ................ 7.13 Seshadri constants 352 ........................ Exercises 355 ................................. 8 Applications of cohomology 359 8.1 The Riemann-Roch theorem 359 .................. Appendix: Residues of differentials on curves 371 ............. 8.2 Comparison of algebraic with analytic cohomology 384 ...... 8.3 de Rham cohomology 406 ...................... Contents vii 8.4 Characteristic p phenomena 412 .. . .................. 8.5 Deformation theory 420 ....................... Exercises 441 ................................. 9 Two deeper results 443 9.1 Mori’s existence theorem of rational curves 443 .......... 9.2 Belyi’s three-point theorem 465 ........... ......... References 475 Index 486 Preface [DM] I gave an introductory course in algebraic geometry many times during the 60’s and 70’s while I was teaching at Harvard. Initially notes to the course mimeographed and bound and sold by the Harvard math department were with a red cover. These old notes werepicked up by Springer and are now sold as the “Red Book of Varieites and Schemes”. However, every time I taught the course, the content changed and grew. I had aimed to eventually publish more polished notes in three volumes. Volume I, dealing with varieties over the complex numbers appeared in 1976and roughly 2/3rds of a nrst draft for volume II was written down at about the same time. This draft covered the material in the Red Book in moredepth and addedsomeadvancedt0pics to give it weight. Volume III was intended to be an introduction to moduli problems but this was never started asmy interests shifted to other neldsin the 80’s. To my surprise, however, some students did read the draft for volume II and felt it made some contribution to the growing literature of multiple introductions to algebraic geometry. The Herculean task of preparing the manuscript for publication, improving and nxing it in multiple ways and adding some half a dozennew sections and results is due to the efforts of the secondauthor. [TO] I had the good fortune of nrst getting acquainted with schemesand functorial approaches in algebraic geometry when the nrst author gavea series of introductory lectures in Tokyo in spring, 1963. Throughout my graduate study at Harvard in the 1960’s, I had many chancesto learn further from the nrst author asmy Ph.D. thesis advisor. It is a great honor and privilege to have this opportunity of sharing with asmany people aspossible the excitement and joy in learning algebraic geometry through the nrst author’s fascinating style. Both authors want to thank those who have assisted in the “penultimate draf ” that we posted on the Web in early 2015, especially Ching—LiChai, Vikraman Balaji, Frans Oort, Fernando Quadros Gouvéa, Amnon Neeman and Akihjko Yukie. A number of extra sections were added to make the book better. Thanks are due to John Tate for the new proof of the Riemann—Rochtheorem, Carlos Simpson for the proof of Belyi’s three-point theorem and Shigefumi Mori for the proofs of some results of his. The expression “added in publication” throughout enables the readers to locate those chapters, sections, propositions, remarks, and footnotes that Preface x were added to the original manuscript of the nrst author. The exercises are those found originally in the manuscript plus further exerciseskindly provided by Ching—LiChai who gave a.graduate course in algebraic geometry at the University of Pennsylvania using a preliminary version of the draft. In 2015, further exercises were provided by Vikraman Balaji (Chennai Mathematical Institute) andD. S.Nagaraj (Institute of Mathematical Sciences, Chennai). Balaji also provided improvements to the “penultimate draft”: the addition of §7.130n Seshadri constants and the introduction to Chapter 9 are due to him. Special thanks are due to Ching—LiChai who in December 2006 drew the second author’s attention to the nrst author’s manuscript, and provided valuable suggestions and improvements during the preparation of this book.
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