Algebraic Geometry

105 Pages·2016·0.872 MB·English

by  Bas Edixhoven, David Holmes and Lenny Taelman

Algebraic Geometry
BasEdixhoven,DavidHolmesandLennyTaelman
January2016
2
Contents
Preface 7
1 Affinespaceandalgebraicsets 9
1.1 TheZariskitopology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 TheNullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 DecompositionofclosedsetsinAn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Application: thetheoremofCayley-Hamilton . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Projectivespaceanditsalgebraicsets 15
2.1 Pnasaset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Pnasatopologicalspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 AmoredirectdescriptionoftheclosedsubsetsofPn . . . . . . . . . . . . . . . . . . . . 18
2.4 HowtoadministratePn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Geometryinprojectivespace 21
3.1 PointsandlinesinP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 CurvesinP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Projectivetransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Affinetransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 Pascal’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Regularfunctionsandalgebraicvarieties 27
4.1 RegularfunctionsonclosedsubsetsofAn . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 RegularfunctionsonclosedsubsetsofPn . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Thecategoryofalgebraicvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Thecategoryofvarieties(continued) 33
5.1 Affinevarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Productsofvarieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Notallcurvescanbeparametrised . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3
4 CONTENTS
6 Presentations,smoothvarietiesandrationalfunctions 39
6.1 Separatedvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.2 Glueingvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.3 Presentationsofvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.4 Smoothvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.5 Rationalfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 Tangentspacesand1-forms 47
7.1 Tangentspacesofembeddedaffinevarieties . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.2 Intrinsicdefinitionofthetangentspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.3 Derivationsanddifferentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.4 Differential1-formsonvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.5 Functionsand1-formsonsmoothirreduciblecurves,ordersandresidues . . . . . . . . . . 50
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
8 ThetheoremofRiemann-Roch 53
8.1 Exactsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.2 Divisorsoncurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.3 H0andH1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.4 TheRiemann-Rochtheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
9 Complexvarietiesandcomplexmanifolds;analytification 61
9.1 Holomorphicfunctionsinseveralvariables . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.2 Complexmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
9.3 Sheavesonabaseforatopology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
9.4 Analytification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
9.4.1 Theunderlyingset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
9.4.2 Thetopology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
9.4.5 TheC-spacestructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
9.4.8 Analytificationofmorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
9.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
9.5.1 Projectiveline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
9.5.2 Affinespace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
10 Riemannsurfaces 69
10.1 Dictionarybetweenvarietiesandmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 69
10.1.1 Projectivevarietiesandcompactness . . . . . . . . . . . . . . . . . . . . . . . . . 69
10.1.4 Furtherproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
10.2 Riemannsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
10.3 Fermatcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
10.3.3 t =(x ,y )withy (cid:54)=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
0 0 0 0
10.3.4 t =(x ,0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
0 0
10.3.5 Pointsatinfinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
10.4 AmapfromtheFermatcurvetoP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
10.5 TriangulationsofRiemannsurfaces,Eulercharacteristic . . . . . . . . . . . . . . . . . . 73
10.6 Genusofatopologicalcover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
10.7 Riemann-Hurwitzformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
CONTENTS 5
10.8 Example: genusofaFermatcurve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
10.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
11 Curvesonsurfaces 77
11.1 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
11.2 Theintersectionpairingonsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
11.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
12 Serreduality,varietiesoverF andtheirzetafunction 83
q
12.1 Serreduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
12.2 ProjectivevarietiesoverF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
q
12.3 DivisorsoncurvesoverF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
q
12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
13 Rationalityandfunctionalequation 89
13.1 Divisorsofgivendegree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
13.2 ThezetafunctionofX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
0
13.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
14 ProofoftheHasse-Weilinequality 93
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
14.2 Self-intersectionofthediagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
14.3 Hodge’sindextheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
14.4 Hasse-Weilinequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
14.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
15 Appendix: ZetafunctionsandtheRiemannhypothesis 99
15.1 TheRiemannzetafunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
15.2 Ringsoffinitetype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
15.3 Zetafunctionsofringsoffinitetype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
15.4 ZetafunctionsofF -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
p
15.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Bibliography 105
Index 105
6 CONTENTS
Preface
This text is based on the lecture notes of the Mastermath course Algebraic Geometry given during the
Springof2009attheUvAbyBasEdixhovenandLennyTaelman. Thosenotesweretyped,asthecourse
wenton,byMichielKosters.Laterversionsincorporatedadditions,correctionsandsuggestionsbyMichiel
Vermeulen,RemkeKloosterman,AriyanJavanpeykar,SamueleAnni,JanRozendaalandRobindeJong.
DavidHolmeshaswrittenLectures9and10whichfocusonaspectsofalgebraicgeometryoverthefield
ofcomplexnumbers. Wethankallthosewhohavecontributed.
The reader will see that this course does not give a systematic introduction to algebraic geometry.
Instead,wehavechosenacleargoal,Andre´ Weil’sproofoftheRiemannhypothesisforcurvesoverfinite
fields using intersection theory on surfaces. Our approach has the advantage that it gets somewhere, but
also the disadvantage that there will be gaps in the exposition, that the reader will have to accept or fill.
Nevertheless,wethinkthatthistextisagoodintroductiontoalgebraicgeometry. Astudentwhowillnot
continue in this matter will have seen beautiful mathematics and learned useful material. A student who
will continue in this area will be motivated for reading the tougher treatments (like [EGA] or the Stacks
project [Stacks]),andwillhaveabiggerchanceofnotgettingstuckintechnicalities. Thiscourseshould
providesufficientbackgroundandmotivationforthestudentwhowantstotakethemoreadvancedcourses
inalgebraicgeometrythatarealsoofferedinMastermath.
Thissyllabusisdividedinto14lectures.Firstthetheoryofalgebraicvarietiesoveralgebraicallyclosed
fieldsisdeveloped,uptotheRiemann-RochtheoremandSerredualityforcurves. Lectures9and10deal
with aspects of varieties when the base field is that of the complex numbers: now, also topological and
analyticaltoolsbecomeavailable. Lectures12–14dealwithvarietiesoverfinitefields(notehoweverthat
finite fields are not algebraically closed!) and lead up to the promised proof of the Riemann hypothesis.
We need one “black box”: Hodge’s Index Theorem. Lecture 11 treats intersection theory on surfaces.
Lecture12introducesthenotionofavarietyoverafinitefield. Lecture13establishestherationalityand
thefunctional equationofthe zetafunctionof curvesoverfinite fields. InLecture14 intersection theory
on surfaces is used to prove the Riemann hypothesis for zeta functions for curves over finite fields. The
Appendixismeanttoprovidemorebackgroundandmotivationonzetafunctions.
Theprerequisitesforthiscoursearethestandardundergraduatealgebracoursesongroups, ringsand
fields (see for example the syllabi [Stev] (in Dutch), or [Lang]), and some basic topology. The reader is
encouragedtocomparethistextwiththeotheralgebraicgeometrysyllabi[Moo]and[Looij].
Nopriorknowledgeofalgebraicgeometryisnecessary.Wewilloccasionallyrefertothebasictextbook
[Hart]onAlgebraicGeometry. Itisstronglyrecommendedtogetholdofthisbook.
7
8 CONTENTS
Lecture 1
Affine space and algebraic sets
InthislecturewewillbasicallytreatSectionI.1of[Hart],i.e.wediscussthemostbasicnotionsofalgebraic
geometry.
Letkbeanalgebraicallyclosedfield. Notethatkisnotafinitefield.
1.1 The Zariski topology
Definition1.1.1 ForninNwedefineaffinen-space,denotedbyAn,askn. ElementsofAnwillbecalled
points. Furthermore,A1iscalledtheaffinelineandA2iscalledtheaffineplane.
Let A = k[x ,...,x ]. We can view an element f of A as a function from An to k by evaluating f at
1 n
pointsofAn.
Definition1.1.2 Wedefinethezerosetofanf inAtobeZ(f) = {p ∈ An : f(p) = 0}. ForS ⊂ Awe
defineZ(S)={p∈An :∀f ∈S, f(p)=0}.
Example1.1.3 LetSbethesubset{x2+x2−1,x }ofk[x ,x ]. ThenZ(S)={(0,1),(0,−1)}⊂A2.
1 2 1 1 2
Remark1.1.4 LetS ⊂ AandI ⊂ AbetheidealgeneratedbyS. ThenZ(I) = Z(S),bythefollowing
argument. SinceS ⊂ I, weobviouslyhaveZ(I) ⊂ Z(S). Ontheotherhand, letp ∈ Z(S)andf ∈ I.
(cid:80) (cid:80)
Writef asafinitesumf = f swithf inA. Thenf(p)= f (p)s(p)=0. HencepisinZ(I).
s∈S s s s s
ThereforewealsohavetheotherinclusionZ(I)⊃Z(S).
Definition1.1.5 AsubsetY ⊂AniscalledalgebraicifthereexistssomeS ⊂AsuchthatY =Z(S).By
thepreviousremarkwecanreplace“someS ⊂A”by“someidealI ⊂A”withoutchangingthemeaning.
Example1.1.6 Weconsiderthecasen = 1. SinceA = k[x]isaprincipalidealdomain(everynon-zero
ideal is generated by its monic element of smallest degree; use division with remainder), the algebraic
subsetsareoftheformZ((f))=Z(f)forsomef inA. Iff =0wegetZ(0)=A1. Iff (cid:54)=0thenf has
onlyafinitenumberofzeros,henceZ(f)isafiniteset. OntheotherhandforeveryfinitesubsetY ofA1
wehaveY =Z((cid:81) (x−p)). ThisshowsthatthealgebraicsubsetsofA1areA1itselftogetherwiththe
p∈Y
finitesubsetsofA1.
Proposition1.1.7 LetnbeinN.
i. LetY ,Y ⊂Anbealgebraicsets. ThenY ∪Y isanalgebraicset.
1 2 1 2
ii. If{Y } isacollectionofalgebraicsubsetsofAn,then∩ Y ⊂Anisalgebraic.
α α α α
9
10 LECTURE1. AFFINESPACEANDALGEBRAICSETS
iii. Anisalgebraic.
iv. ∅isalgebraic.
Proof i. WeclaimthatforS andS subsetsofAwehave
1 2
Z(S )∪Z(S )=Z(S S ), where S S ={fg :f ∈S , g ∈S }.
1 2 1 2 1 2 1 2
Obviously we have Z(S ) ∪ Z(S ) ⊂ Z(S S ). For the other inclusion, assume that p ∈ Z(S S )
1 2 1 2 1 2
and p (cid:54)∈ Z(S ). Then there is an f in S such that f(p) (cid:54)= 0. But we have for all g in S that
1 1 2
0=(fg)(p)=f(p)g(p). Sincef(p)(cid:54)=0,wegetthatg(p)=0forallg ∈S ,andhencep∈Z(S ).
2 2
ii. WeobviouslyhaveZ(∪ S )=∩ Z(S ).
α α α α
iii. NotethatZ(∅)=Z(0)=A1.
iv. NotethatZ(1)=∅. (cid:3)
Corollary1.1.8 ThealgebraicsubsetsofAnarethe closedsubsetsofatopologyonAn. Wewillcallthis
topologythe Zariskitopology.
Remark1.1.9 ByExample1.1.6theZariskitopologyonA1isequaltotheco-finitetopologyonA1.
Definition1.1.10 OnasubsetY ⊂ An wedefinetheZariskitopologyastheinducedtopologyfromthe
ZariskitopologyonAn.
Definition1.1.11 AtopologicalspaceX isirreducibleif(1)X (cid:54)=∅and(2)ifX =Z ∪Z withZ and
1 2 1
Z closedsubsetsofX thenZ =X orZ =X.
2 1 2
Examples1.1.12 The affine line A1 is irreducible, since A1 is infinite. The real line R with its usual
topologyisnotirreducible,becauseR=(−∞,0]∪[0,∞).
Remark1.1.13 Let Y be a non-empty subset of An. Then Y is irreducible if and only if for all closed
subsetsZ andZ ofAnwithY ⊂Z ∪Z onehasY ⊂Z orY ⊂Z .
1 2 1 2 1 2
1.2 The Nullstellensatz
Let n be in N and A = k[x ,...,x ]. Recall that k is an algebraically closed field. In the previous
1 n
subsectionwedefinedamap:
Z :{subsetsofA}→{closedsubsetsofAn}.
Wewouldliketo“invert”thismapZ. NotethatZissurjective,butnotinjective,notevenwhenwerestrict
to the set of ideals: for example Z((x)) = Z((x2)) ⊂ A1. The problem comes from the fact that if
fm(p)=0forsomef ∈Aandm≥1thenf(p)=0aswell.
Definition1.2.1 Wedefinethefollowingmap:
I :{subsetsofAn} → {idealsofA}
Y (cid:55)→ I(Y):={f ∈A:∀p∈Y, f(p)=0}
Definition1.2.2 AringRisreduced ifitsonlynilpotentelementis0(examples: integraldomain,fields,
productsofintegraldomains,subringsofreducedrings). AnidealI inaringRisradicalifforallainR
andminZ suchthatam ∈I,aisinI.
≥1