Algebraic geometry: an introduction

266 Pages·2008·1.76 MB·English

by  Perrin D.

Daniel Perrin
Algebraic Geometry
An Introduction
Translated from the French by
Catriona Maclean
ProfessorDanielPerrin
De´partementdeMathe´matiques
Bˆat.425
Faculte´desSciencesd’Orsay
Universite´Paris-Sud11
F-91405OrsayCedex
France
Translator:
CatrionaMaclean
InstitutFourier
UFRdeMathe´matiques
UMR5582CNRS/Universite´J.Fourier
100ruedesMaths,B.P.74
F-38402StMartind’He`resCedex
France
ISBN:978-1-84800-055-1 e-ISBN:978-1-84800-056-8
DOI:10.1007/978-1-84800-056-8
BritishLibraryCataloguinginPublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressControlNumber:2007935214
MathematicsSubjectClassification(2000):14-01,14Axx,14H50,14M06
EDPSciencesISBN978-2-7598-0048-3
TranslationfromtheFrenchlanguageedition:
Ge´ome´triealge´briquebyDanielPerrin
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Contents
Preface ........................................................ IX
Notation....................................................... XI
Introduction................................................... 1
0 Algebraic geometry ..................................... 1
1 Some objects .......................................... 1
2 Some problems......................................... 4
I Affine algebraic sets ...................................... 9
1 Affine algebraic sets and the Zariski topology .............. 9
2 Ideal of an affine algebraic set............................ 11
3 Irreducibility........................................... 13
4 The Nullstellensatz (or Hilbert’s zeros theorem) ............ 15
5 A first step towards B´ezout’s theorem..................... 19
6 An introduction to morphisms ........................... 20
Exercises .................................................. 24
II Projective algebraic sets .................................. 25
0 Motivation ............................................ 25
1 Projective space........................................ 25
2 Homographies.......................................... 27
3 Relation between affine and projective space ............... 27
4 Projective algebraic sets................................. 29
5 Ideal of a projective algebraic set ......................... 31
6 A graded ring associated to a projective algebraic set ....... 32
7 Appendix: graded rings ................................. 33
Exercises .................................................. 34
VI Contents
III Sheaves and varieties ..................................... 37
0 Motivation ............................................ 37
1 The sheaf concept ...................................... 38
2 The structural sheaf of an affine algebraic set .............. 41
3 Affine varieties ......................................... 43
4 Algebraic varieties...................................... 44
5 Local rings ............................................ 47
6 Sheaves of modules ..................................... 48
7 Sheaves of modules on an affine algebraic variety ........... 50
8 Projective varieties ..................................... 52
9 Sheaves of modules on projective algebraic varieties......... 56
10 Two important exact sequences .......................... 59
11 Examples of morphisms ................................. 60
Exercises A ................................................ 63
Exercises B ................................................ 66
IV Dimension................................................ 69
0 Introduction ........................................... 69
1 The topological definition and the link with algebra......... 69
2 Dimension and counting equations........................ 72
3 Morphisms and dimension ............................... 77
4 Annex: finite morphisms................................. 82
Exercises .................................................. 83
V Tangent spaces and singular points ....................... 87
0 Introduction ........................................... 87
1 Tangent spaces......................................... 88
2 Singular points......................................... 91
3 Regular local rings...................................... 93
4 Curves ................................................ 95
Exercises .................................................. 97
VI B´ezout’s theorem.........................................101
0 Introduction ...........................................101
1 Intersection multiplicities................................101
2 B´ezout’s theorem.......................................106
Exercises ..................................................111
VII Sheaf cohomology ........................................113
0 Introduction ...........................................113
1 Some homological algebra ...............................114
2 Cˇech cohomology.......................................117
3 Vanishing theorems.....................................121
4 The cohomology of the sheaves O (d)....................122
Pn
Exercises ..................................................127
Contents VII
VIIIArithmetic genus of curves and the weak Riemann-Roch
theorem ..................................................131
0 Introduction: the Euler-Poincar´e characteristic .............131
1 Degree and genus of projective curves, Riemann-Roch 1 .....132
2 Divisors on a curve and Riemann-Roch 2 ..................138
Exercises ..................................................147
IX Rational maps, geometric genus and rational curves ......149
0 Introduction ...........................................149
1 Rational maps .........................................149
2 Curves ................................................152
3 Normalisation: the algebraic method ......................155
4 Affine blow-ups ........................................158
5 Global blow-ups........................................163
6 Appendix: review of the above proofs .....................170
X Liaison of space curves ...................................173
0 Introduction ...........................................173
1 Ideals and resolutions ...................................174
2 ACM curves ...........................................179
3 Liaison of space curves ..................................187
Exercises ..................................................194
Appendices
A Summary of useful results from algebra...................199
1 Rings .................................................199
2 Tensor products........................................204
3 Transcendence bases ....................................206
4 Some algebra exercises ..................................207
B Schemes ..................................................209
0 Introduction ...........................................209
1 Affine schemes .........................................210
2 Schemes...............................................210
3 What changes when we work with schemes ................211
4 Why working with schemes is useful ......................212
5 A scheme-theoretic Bertini theorem.......................213
C Problems .................................................215
Problem I..................................................215
Problem II .................................................217
Problem III ................................................218
Problem IV ................................................220
Problem V .................................................222
VIII Contents
Problem VI ................................................223
Problem VII ...............................................225
Problem VIII...............................................228
Problem IX ................................................230
Midterm, December 1991 ....................................232
Exam, January 1992 ........................................234
Exam, June 1992 ...........................................238
Exam, January 1993 ........................................239
Exam, June 1993 ...........................................242
Exam, February 1994........................................245
References.....................................................251
Index of notations .............................................253
Index..........................................................255
Preface
This book is built upon a basic second-year masters course given in 1991–
1992, 1992–1993 and 1993–1994 at the Universit´e Paris-Sud (Orsay). The
course consisted of about 50 hours of classroom time, of which three-quarters
werelecturesandone-quarterexamplesclasses.Itwasaimedatstudentswho
had no previous experience with algebraic geometry. Of course, in the time
available, it was impossible to cover more than a small part of this field. I
chose to focus on projective algebraic geometry over an algebraically closed
base field, using algebraic methods only.
The basic principles of this course were as follows:
1) Start with easily formulated problems with non-trivial solutions (such
as B´ezout’s theorem on intersections of plane curves and the problem of
rationalcurves).In1993–1994,thechapteronrationalcurveswasreplaced
by the chapter on space curves.
2) Usetheseproblemstointroducethefundamentaltoolsofalgebraicgeom-
etry: dimension, singularities, sheaves, varieties and cohomology. I chose
not to explain the scheme-theoretic method other than for finite schemes
(which are needed to be able to talk about intersection multiplicities). A
short summary is given in an appendix, in which special importance is
given to the presence of nilpotent elements.
3) Use as little commutative algebra as possible by quoting without proof
(or proving only in special cases) a certain number of theorems whose
proof is not necessary in practise. The main theorems used are collected
in a summary of results from algebra with references. Some of them are
suggested as exercises or problems.
4) Do not hesitate to quote certain algebraic geometry theorems when the
proof’s absence does not alter the reader’s understanding of the result.
For example, this is the case for the uniqueness of cohomology or certain
technicalpointsinChapterIX.Moregenerally,inwritingthisbookItried
to privilege understanding of phenomena over technique.
X Preface
5) Provide a certain number of exercises and problems for every subject
discussed. The papers of all the exams for this course are given in an
appendix at the end of the book.
Clearly, a book on this subject cannot pretend to be original. This work
is therefore largely based on existing works, particularly the books by
Hartshorne [H], Fulton [F], Mumford [M] and Shafarevitch [Sh].
I would like to thank Mireille Martin-Deschamps for her careful read-
ing and her remarks. I would also like to thank all those who attended the
course and who pointed out several errors and suggested improvements, no-
tablyAbdelkaderBelkilani,NicusorDan,LeopoldoKulesz,VincentLafforgue
and Thomas P´eteul.
I warmly thank Catriona Maclean for her careful translation of this book
into English.
Andfinally,itismypleasuretothankClaudeSabbahforhavingaccepted
the French edition of this book in the series Savoirs Actuels and for his help
with the editing of the final English edition.
Notation
We denote the set of positive integers (resp.the set of integers, rational num-
bers, real numbers or complex numbers) by N (resp.Z, Q, R or C). We
denote by F the finite field with q elements.
q
We denote the cardinal of a set(cid:1)E(cid:2)by |E|. We denote the integral part of
a real number by [x]. The notation n represents the binomial coefficient:
p
(cid:3) (cid:4)
n n!
= .
p p!(n−p)!
By convention, this coefficient is zero whenever n<p.
If f :G→H is a homomorphism of abelian groups (or modules or vector
spaces),wewilldenotebyKerf (resp.Imf,resp.Cokerf)itskernel(resp.its
image, resp.its cokernel). We recall that by definition Cokerf =H/Imf.
An exact sequence of abelian groups (or modules or vector spaces)
0−→M(cid:1) −−u→M −−v→M(cid:1)(cid:1) −→0
is given by the data of two homomorphisms u,v such that
a) u is injective,
b) v is surjective,
c) Imu=Kerv.
Further definitions and notations are contained in the summary of useful
results from algebra.
In the exercises and problems, the symbol ¶ indicates a difficult question.