作者：R.Hartshorne
页数：496
出版社：世界图书出版公司
出版日期：1999
ISBN：9787506200820
电子书格式：pdf/epub/txt
内容简介
片断：
CHAPTER1
Varieties
Ourpurposeinthischapteristogiveanintroductiontoalgebraicgeometry
withaslittlemachineryaspossible.Weworkoverafixedalgebraically
closedfieldk.Wedefinethemainobjectsofstudy,whicharealgebraic
varietiesinaffineorprojectivespace.Weintroducesomeofthemost
importantconcepts,suchasdimension,regularfunctions,rationalmaps,
nonsingularvarieties,andthedegreeofaprojectivevariety.Andmostim-
portant.wegivelotsofspecificexamples,intheformofexercisesattheend
ofeachsection.Theexampleshavebeenselectedtoillustratemanyinter-
estingandimportantphenomena,beyondthosementionedinthetext.The
personwhostudiestheseexamplescarefullywillnotonlyhaveagoodunder-
standingofthebasicconceptsofalgebraicgeometry.buthewillalsohave
thebackgroundtoappreciatesomeofthemoreabstractdevelopmentsof
modernalgebraicgeometry,andhewillhavearesourceagainstwhichto
checkhisintuition.Wewillcontinuallyreferbacktothislibraryofexamples
intherestofthebook.
Thelastsectionofthischapterisakindofsecondintroductiontothebook.
Itcontainsadiscussionofthe”classificationproblem,”whichhasmotivated
muchofthedevelopmentofalgebraicgeometry.Italsocontainsadiscussion
ofthedegreeofgeneralityinwhichoneshoulddevelopthefoundationsof
algebraicgeometry,andassuchprovidesmotivationforthetheoryof
schemes.
作者简介
世界图书出版公司成立于1986年，是一家主要从事版权贸易的出版集团公司，在北京、上海、广州、西安设有四家分公司。 世界图书出版公司通过版权贸易，获得海外出版机构的授权，重印、加注中文、翻译出版或同海外出版机构联合出版各类图书及配套磁带，同时也出版中国作者撰写的科学技术、社会科学、语言学习及工具书。
本书特色
This book provides an introduction to abstract algebraic geometry using the methods of schemes and cohomology. The main objects of study are algebraic varieties in an affine or projective space over an algebraically closed field; these are introduced in Chapter I, to establish a number of basic concepts and examples. Then the methods of schemes and cohomology are developed in Chapters II and III, with emphasis on applications rather than excessive generality. The last two chapters of the book (IV and V) use these methods to study topics in the classical theory of algebraic curves and surfaces.
本书为英文版。
目录
CHAPTER I Varieties
1 Affine Varieties
2 Projective Varieties
3 Morphisms
4 Rational Maps
5 Nonsingular Varieties
6 Nonsingular Curves
7 Intersections in Projective Space
8 What Is Algebraic Geometry?
CHAPTER II Schemes
1 Sheaves
2 Schemes
3 First Properties of Schemes
4 Separated and Proper Morphisms
5 Sheaves of Modules
6 Divisors
7 Projective Morphisms
8 Differentials
9 Formal Schemes
CHAPTER III Cohomology
1 Derived Functors
2 Cohomology of Sheaves
3 Cohomology of a Noetherian Affine Scheme
4 Cech Cohomology
5 The Cohomology of Projective Space
6 Ext Groups and Sheaves
7 The Serre Duality Theorem
8 Higher Direct images of Sheaves
9 Flat Morphisms
10 Smooth Morphisms
11 The Theorem on Formal Functions
12 The Semicontinuity Theorem
CHAPTER IV Curves
1 Riemann-Roch Theorem
2 Hurwitz’s Theorem
3 Embeddings in Projective Space
4 Elliptic Carves
5 The Canonical Embedding
6 Classification of Curves in P3
CHAPTER V Surfaces
1 Geometry on a Surface
2 Ruled Surfaces
3 Monoidal Transformations
4 The Cubic Surface in P3
5 Birational Transformations
6 Classification of Surfaces
APPENDIX A
APPENDIX B
APPENDIX C
Bibliography
Results from Algebra
Glossary of Notations
Index
节选
This book provides an introduction to abstract algebraic geometry using the methods of schemes and cohomology. The main objects of study are algebraic varieties in an affine or projective space over an algebraically closed field; these are introduced in Chapter I, to establish a number of basic concepts and examples. Then the methods of schemes and cohomology are developed in Chapters II and III, with emphasis on applications rather than excessive generality. The last two chapters of the book (IV and V) use these methods to study topics in the classical theory of algebraic curves and surfaces.
