Undergraduate algebraic geometry [Lecture notes]

162 Pages·2016·2.812 MB·English

by  Joe Harris, A. Landesman

MATH 137 NOTES: UNDERGRADUATE ALGEBRAIC
GEOMETRY
AARONLANDESMAN
CONTENTS
1. Introduction 6
2. Conventions 6
3. 1/25/16 7
3.1. Logistics 7
3.2. HistoryofAlgebraicGeometry 7
3.3. AncientHistory 7
3.4. TwentiethCentury 10
3.5. Howthecoursewillproceed 10
3.6. Beginningofthemathematicalportionofthecourse 11
4. 1/27/16 13
4.1. LogisticsandReview 13
5. 1/29/16 17
5.1. Logisticsandreview 17
5.2. TwistedCubics 17
5.3. BasicDefinitions 21
5.4. Regularfunctions 21
6. 2/1/16 22
6.1. LogisticsandReview 22
6.2. TheCategoryofAffineVarieties 22
6.3. TheCategoryofQuasi-ProjectiveVarieties 24
7. 2/4/16 25
7.1. Logisticsandreview 25
7.2. MoreonRegularMaps 26
7.3. VeroneseMaps 27
7.4. SegreMaps 29
8. 2/5/16 30
8.1. OverviewandReview 30
8.2. MoreontheVeroneseVariety 30
8.3. MoreontheSegreMap 32
9. 2/8/16 33
9.1. Review 33
1
2 AARONLANDESMAN
9.2. EvenMoreonSegreVarieties 34
9.3. Cones 35
9.4. ConesandQuadrics 37
10. 2/10/15 38
10.1. Review 38
10.2. MoreonQuadrics 38
10.3. ProjectionAwayfromaPoint 39
10.4. Resultants 40
11. 2/12/16 42
11.1. Logistics 42
11.2. Review 43
11.3. ProjectionofaVarietyisaVariety 44
11.4. Anyconeoveravarietyisavariety 44
11.5. ProjectionofaVarietyfromaProductisaVariety 45
11.6. TheimageofaregularmapisaVariety 45
12. 2/17/16 46
12.1. Overview 46
12.2. Families 47
12.3. UniversalFamilies 47
12.4. SectionsofUniversalFamilies 49
12.5. MoreexamplesofFamilies 50
13. 2/19/16 51
13.1. Review 51
13.2. Generality 51
13.3. IntroductiontotheNullstellensatz 54
14. 2/22/16 55
14.1. ReviewandOverviewofComingAttractions 55
14.2. Whatscissorsaregoodfor: Cuttingout 56
14.3. Irreducibility 58
15. 2/24/16 59
15.1. Overview 59
15.2. Preliminaries 59
15.3. PrimaryDecomposition 60
15.4. TheNullstellensatz 61
16. 2/26/15 63
16.1. ReviewoftheNullstellensatz 63
16.2. Preliminaries to Completing the proof of the
Nullstellensatz 63
16.3. ProofofNullstellensatz 64
16.4. Grassmannians 65
17. 2/29/16 66
17.1. Overviewandahomeworkproblem 66
MATH137NOTES:UNDERGRADUATEALGEBRAICGEOMETRY 3
17.2. Multilinearalgebra,intheserviceofGrassmannians 68
17.3. Grassmannians 70
18. 3/2/16 70
18.1. Review 70
18.2. Thegrassmannianisaprojectivevariety 71
18.3. Playingwithgrassmannian 73
19. 3/4/16 74
19.1. Reviewoflasttime: 74
19.2. FindingtheequationsofG(1,3) 75
19.3. SubvarietiesofG(1,3) 77
19.4. Answeringourenumerativequestion 78
20. 3/7/16 79
20.1. Plan 79
20.2. Rationalfunctions 79
21. 3/9/16 82
21.1. RationalMaps 82
21.2. Operationswithrationalmaps 82
22. 3/11/16 85
22.1. Calculus 85
P2
22.2. Blowupof atapoint 86
22.3. Blowupsingeneral 87
22.4. Example: joinofavariety 88
23. 3/21/16 88
23.1. Secondhalfofthecourse 88
23.2. Dimensionhistorically 89
23.3. Usefulcharacterizationofdimension 90
23.4. Theofficialdefinition 91
23.5. Non-irreducible,non-projectivecase 91
24. 3/23/16 92
24.1. Homework 92
24.2. Calculatingdimension 92
24.3. DimensionoftheGrassmannian 94
24.4. Dimensionoftheuniversalk-plane 94
24.5. Dimensionofthevarietyofincidentplanes 95
25. 3/25/16 96
25.1. Review 96
25.2. Incidencevarieties 97
25.3. AnsweringQuestion25.1 98
26. 3/28/16 99
26.1. SecantVarieties 99
26.2. DeficientVarieties 100
27. 3/30/16 103
4 AARONLANDESMAN
27.1. Schedule 103
27.2. Thelocusofmatricesofagivenrank 104
27.3. Polynomialsasdeterminants 106
28. 4/1/16 108
28.1. Provingthemaintheoremofdimensiontheory 108
28.2. Funwithdimensioncounts 110
29. 4/4/16 113
29.1. Overview 113
29.2. Hilbertfunctions 113
30. 4/6/16 118
30.1. Overviewandreview 118
30.2. An alternate proof of Theorem 30.1, using the Hilbert
syzygytheorem 119
31. 3/8/16 121
31.1. Minimalresolutionofthetwistedcubic 121
31.2. Tangentspacesandsmoothness 124
32. 4/11/16 126
32.1. Overview 126
32.2. Definitionsoftangentspaces 126
32.3. Constructionswiththetangentspace 128
33. 4/13/16 130
33.1. Overview 130
33.2. DualVarieties 130
33.3. ResolutionofSingularities 132
33.4. NashBlowUps 132
34. 4/15/16 134
34.1. Bertini’sTheorem 134
34.2. TheLefschetzprinciple 135
34.3. Degree 137
35. 4/18/16 139
35.1. Review 139
35.2. Bezout’sTheorem,partI:Asimplerstatement 140
35.3. ExamplesofBezout,partI 141
36. 4/20/16 143
36.1. Overview 143
36.2. Moredegreecalculations 143
37. ProofofBezout’sTheorem 145
38. 4/22/16 147
38.1. Overview 147
38.2. Binomialidentities 147
38.3. Reviewofthesetupfromthepreviousclass 148
38.4. ProvingBezout’stheorem 149
MATH137NOTES:UNDERGRADUATEALGEBRAICGEOMETRY 5
38.5. StrongBezout 150
39. 4/25/16 152
39.1. Overview 152
39.2. Thequestionofconnectedcomponentsoverthereals 153
39.3. Thegenusisconstantinfamilies 153
39.4. Findingthegenus 155
R
39.5. Findingthenumberofconnectedcomponentsof 156
40. 4/27/16 158
40.1. Finishingupwiththenumberofconnectedcomponents
ofrealplanecurves 158
40.2. Parameterspaces 158
40.3. SuccessfulAttempt4 161
6 AARONLANDESMAN
1. INTRODUCTION
JoeHarristaughtacourse(Math137)onundergraduatealgebraic
geometryatHarvardinSpring2016.
These are my “live-TEXed“ notes from the course. Conventions
are as follows: Each lecture gets its own “chapter,” and appears in
thetableofcontentswiththedate.
Ofcourse,thesenotesarenotafaithfulrepresentationofthecourse,
either in the mathematics itself or in the quotes, jokes, and philo-
sophical musings; in particular, the errors are my fault. By the same
token, any virtues in the notes are to be credited to the lecturer and
not the scribe. 1 Several of the classes have notes taken by Hannah
Larson. ThankstoherfortakingnoteswhenImissedclass.
[email protected].
2. CONVENTIONS
Herearesomeconventionswewilladaptthroughoutthenotes.
(1) I have a preference toward making any detail stated in class,
which is not verified, into an exercise. This will mean there
are many trivial exercises in the notes. When the exercise
seems nontrivial, I will try to give a hint. Feel free to contact
me if there are any exercises you do not know how to solve
orotherdetailswhichareunclear.
(2) Throughout the notes, I will often include parenthetical re-
marks which describe things beyond the scope of the course.
If some word or explanation is placed in quotation marks
or in parentheses, and you don’t understand it, don’t worry
about it! It’s more meant to give you a flavor of how one
might describe the 19th century ideas in this course in terms
of20thcenturyalgebraicgeometry.
1ThisintroductionhasbeenadaptedfromAkhilMatthew’sintroductiontohis
notes,withhispermission.
MATH137NOTES:UNDERGRADUATEALGEBRAICGEOMETRY 7
3. 1/25/16
3.1. Logistics.
(1) ThreelecturesaweekonMonday,Wednesday,andFriday
(2) We’llhaveweeklyrecitations
(3) This week we’ll have special recitations on Wednesday and
Fridayat3pm;
(4) You’rewelcometogotoeithersection
(5) Aaronwillbetakingnotes
(6) PleasegivefeedbacktoProfessorHarrisortheCAs
(7) We’ll have weekly problem sets, due Fridays. The first prob-
lemsetmightbeduenextWednesdayorso.
(8) TheminimalprerequisiteisMath122.
(9) ThemaintextisJoeHarris’“Afirstcourseinalgebraicgeom-
etry.”
(10) Thebookgoesfastinmanyrespects. Inparticular,thedefini-
tionofprojectivespaceisgivenlittleattention.
(11) Somepartsofothermathwillbeused. Forexample,knowing
about topology or complex analysis will be useful to know,
butwe’lldefineeverytermweuse.
3.2. HistoryofAlgebraicGeometry. Today,wewillgothroughthe
Historyofalgebraicgeometry.
3.3. Ancient History. The origins of algebraic geometry are in un-
derstanding solutions of polynomial equations. People first stud-
ied this around the late middle ages and early renaissance. People
started by looking at a polynomial in one variables f(x). You might
thenwanttoknow:
Question3.1. Whatarethesolutionstof(x) = 0?
(1) If f(x) is a quadratic polynomial, you can use the quadratic
formula.
(2) For some time, people didn’t know how to solve cubic poly-
nomials. But, after some time, people found an explicit for-
mulaforcubics.
(3) Inthebeginningofthe19thcentury,usingGaloistheory,peo-
ple found this pattern does not continue past degree 5, and
therearenoclosedformsolutionsindegree5ormore.
So,peoplejumpedtothenextlevelofcomplexity: polynomialsin
twovariables.
8 AARONLANDESMAN
Question3.2. Whatarethesolutionstof(x,y) = 0? Or,whatarethe
simultaneoussolutionstoacollectionofpolynomialsf (x,y) = 0for
i
alli?
Algebraicgeometrybeginshere.
Goal 3.3. The goal of algebraic geometry is to relate the algebra of f
tothegeometryofitszerolocus.
This was the goal until the second decade of the nineteenth cen-
tury. At this point, two fundamental changes occurred in the study
ofthesubject.
3.3.1. Nineteenthcentury. In1810,Ponceletmadetwobreakthroughs.
Around this time, Poncelet was captured by the Russians, and held
asaprisoner. Inthecourseofhiscaptivity,hefoundtwofundamen-
talchanges.
(1) Workoverthecomplexnumbersinsteadofthereals.
(2) Work in projective space (to be described soon), instead of
Cn
affinespace,thatis,in .
There are several reasons for these changes. Let’s go back to the
one variable case. The reason the complex numbers are nice is the
following:
Theorem3.4(FundamentalTheoremofAlgebra). Apolynomialofde-
greenalwayshasnsolutionsoverthecomplexnumbers.
Example 3.5. This is false over the real numbers. For example, con-
siderf(x) = x2+1.
This makes things nicer over the complex numbers. If we have
a family of polynomial equations (meaning that we vary the coeffi-
cients) then the number of solutions will be constant over the com-
plexnumbers.
Let’sseetwoexamplesofwhatcangowrong,andhowPoncelet’s
two changes help this. The motivation is to preserve the number of
intersectionofsomealgebraicvarieties.
(1)
Example 3.6. Suppose we have a line and a conic (e.g., a cir-
cle,orsomethingcutoutbyadegree2polynomialinthepro-
jective plane). If the line meets conic, over the real numbers,
there will be0, 1, or 2 solutions, depending on where theline
liesrelativetothecircle.
(2)
MATH137NOTES:UNDERGRADUATEALGEBRAICGEOMETRY 9
FIGURE 1. As the lines limit away from the ellipse
overtherealnumbers,theystartwithtwointersection
points, limit to 1, and then pass to not intersect at all.
Thissituationisrectifiedwhenweworkoverthecom-
plex numbers (although we will have to “count with
multiplicity”).
Example 3.7. If we have two lines which are parallel, they
won’tmeet. But,iftheyarenotparallelwillmeet. So,weneed
to add a “point at infinity.” This motivates using projective
space,whichdoesincludethesepoints.
Remark 3.8. By working over the complex numbers and in projec-
tive space, you lose the ability to visualize some of these phenom-
ena. However, you gain a lot. For example, heuristically speaking,
you obtain that the “number of solutions” will be constant in nice
(flat)families.
Let’sseeanexample:
Example3.9. Considerx2+y2 andx2−y2,whichlookverydifferent
overthereals. Thefirstonlyhasazeroattheorigin,whilethesecond
is two lines. However, over the complex numbers, they are related
byachangeofvariablesy (cid:55)→ iy.
Infact,inprojectivespace,thehyperbola,parabola,andellipseare
essentially the same. The only difference is how they meet the line
at infinity over the real numbers. (Over the complex numbers, they
10 AARONLANDESMAN
FIGURE 2. As the blue lines limit to the left black line,
their point of intersection limits down the right black
line to the point at infinity. Hence, we need to include
thispointatinfinitysothattwoparallellineswillmeet.
will all meet the line at infinity “with multiplicity two.”) Over the
reals, a parabola is tangent to the line at infinity, a hyperbola meets
thelineatinfinitytwice,andanellipsedoesn’tmeetit.
3.4. Twentieth Century. In 1900 to 1950, there was an algebraiciza-
tionofalgebraicgeometry,byZariskiandWeil.
Between1950andthepresent,therewasthedevelopmentofwhat
we think of as modern algebraic geometry. There was in introduc-
tionofschemes,sheaves,stacks,etc. Thiswasdue,amongothers,to
Grothendieck,Serre,Mumford,Artin,etc.
Remark 3.10. Once again, this led to a trade off. This led to a theory
with much greater power. However, the downside is that you lose a
lot of intuition which you used to have over the complex numbers.
If one wants to do algebraic geometry, one does need to learn the
modern version. However, jumping in and trying to learn this, is
likebeatingyourheadagainstthewall.
3.5. How the course will proceed. We will stick to the 19th century
version of algebraic geometry in this course. We won’t shy away
fromusingthingslikeringsandfields,butwewillnotuseanydeep
theoremsfromcommutativealgebra.