An Undergraduate Primer in Algebraic Geometry

323 Pages·2021·4.901 MB·English

by  Ciro Ciliberto

UNITEXT 129
Ciro Ciliberto
An Undergraduate
Primer in Algebraic
Geometry
UNITEXT - La Matematica per il 3+2
Volume 129
Editor-in-Chief
AlfioQuarteroni,PolitecnicodiMilano,Milan,Italy;ÉcolePolytechniqueFédérale
de Lausanne (EPFL), Lausanne, Switzerland
Series Editors
Luigi Ambrosio, Scuola Normale Superiore, Pisa, Italy
Paolo Biscari, Politecnico di Milano, Milan, Italy
Ciro Ciliberto, Università di Roma “Tor Vergata”, Rome, Italy
Camillo De Lellis, Institute for Advanced Study, Princeton, NJ, USA
Massimiliano Gubinelli, Hausdorff Center for Mathematics, Rheinische
Friedrich-Wilhelms-Universität, Bonn, Germany
Victor Panaretos, Institute of Mathematics, EPFL, Lausanne, Switzerland
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Ciro Ciliberto
An Undergraduate Primer
in Algebraic Geometry
123
Ciro Ciliberto
Dipartimento di Matematica
Universitàdi Roma“Tor Vergata”
Rome, Italy
ISSN 2038-5714 ISSN 2532-3318 (electronic)
UNITEXT- La Matematica peril3+2
ISSN 2038-5722 ISSN 2038-5757 (electronic)
ISBN978-3-030-71020-0 ISBN978-3-030-71021-7 (eBook)
https://doi.org/10.1007/978-3-030-71021-7
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Preface
This book arouse from notes of courses in Algebraic Geometry that I gave at the
University of Roma Tor Vergata during several years. It basically consists of two
parts.ThefirstincludesChapters1–14,anditisdevotedtoanintroductiontobasic
conceptsinAlgebraicGeometry.Themainobjectsofinterestinthispartareaffine
and projective varieties, some of their main attributes (like irreducibility, dimen-
sion, regular and rational functions, morphisms, products, degree, etc.) and basic
examples (hypersurfaces, Veronese varieties, Segre varieties, blow-ups,
Grassmannians, etc.). One of the leading themes in this first part is elimination
theory,towhichseveralsectionsofthebookarededicated,andwhichisatthebasis
of some important applications like Hilbert’s Nullstellensatz and basic intersection
theory of varieties with Bezout’s Theorem. The second part of the book, which
includesChaps.15–20,isdevotedtothetheoryofcurves.Abasicpreliminaryisin
Chap. 15 with formalpowerseries,whichconstitute themain tool for the study of
localpropertiesofcurves.ThenChaps.16and17aredevotedtothestudyofaffine
and projective plane curves, respectively. Chapter 18 contains the proof of reso-
lution of singularities of curves. Chapter 19 is devoted to the classical theory of
linear equivalence of divisors and linear series on a curve. Finally, Chap. 20 con-
tains the Riemann–Roch and Riemann–Hurwitz Theorems.
The approach in this book is purely algebraic, so no analysis or differential
geometryisneeded.Themaintooliscommutativealgebrafromwhichwerecallthe
main results we need, in most cases with proofs. The prerequisites consist in the
knowledge of basics in affine and projective geometry (in particular, conics and
quadrics in three-dimensional space), basic algebraic concepts regarding rings,
modules,fields, linear algebra and basic notionsin thetheory ofcategories.A few
elementary facts of topology are needed in Chap. 4.
The book can be used as a textbook for a basic undergraduate course in
AlgebraicGeometry.Theusersofthebookarenotnecessarilyintendedtobecome
algebraicgeometersbutmaybesimplyinterestedstudentsorresearcherswhowant
tohaveafirstsmatteringinthetopic.Chapter14isnotessentialfortherestandcan
be skipped in a first reading. For a short introductory course, one can focus on the
first thirteen chapters only.
v
vi Preface
Thebookcontainsseveralexercises,inwhichtherearemoreexamplesandparts
ofthetheorywhicharenotfullydevelopedinthetext.Someexercisesaremarked
with an asterisk, which means either that they are a bit more difficult than the
average,orthattheyareneededforthesequelofthebook.Ofsomeexercises,there
are the solutions at the end of each chapter.
Whatreaderswillnotfindinthisbookare(atleast)twomainthings.Thefirstis
sheaf theory, cohomology, schemes, etc. For this, the classical references are [4]
and, in part, the second volume of [7]. The second is a computational approach to
Algebraic Geometry, which is a very interesting topic for which I recommend [2].
I am indebted for inspiration to several sources, for instance [3, 6, 7, 8].
Rome, Italy Ciro Ciliberto
October 2020
Contents
1 Affine and Projective Algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Affine Algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Graded Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Projective Algebraic Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Projective Closure of Affine Sets . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6.1 Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6.2 Projective Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6.3 Affine Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6.4 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6.5 Divisors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6.6 Product Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Basic Notions of Elimination Theory and Applications. . . . . . . . . . 25
2.1 The Resultant of Two Polynomials . . . . . . . . . . . . . . . . . . . . . 25
2.2 The Intersection of Two Plane Curves . . . . . . . . . . . . . . . . . . . 28
2.3 Kronecker Elimination Method: One Variable . . . . . . . . . . . . . 30
2.4 Kronecker Elimination Method: More Variables. . . . . . . . . . . . 32
2.5 Hilbert Nullstellensatz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Zariski Closed Subsets and Ideals in the Polynomials Ring . . . . . . 39
3.1 Ideals and Coordinate Rings . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Maximal Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.2 The Twisted Cubic. . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.3 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 43
vii
viii Contents
4 Some Topological Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Irreducible Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Noetherian Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Topological Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Regular and Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1 Regular Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3 Local Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.4 Integral Elements over a Ring . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.5 Subvarieties and Their Local Rings . . . . . . . . . . . . . . . . . . . . . 59
5.6 Product of Affine Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.7 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.1 The Definition of Morphism . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Which Maps Are Morphisms. . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 Affine Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.4 The Veronese Morphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.5 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7 Rational Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.1 Definition of Rational Maps and Basic Properties. . . . . . . . . . . 85
7.2 Birational Models of Quasi-projective Varieties . . . . . . . . . . . . 91
7.3 Unirational and Rational Varieties . . . . . . . . . . . . . . . . . . . . . . 93
7.4 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8 Product of Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.1 Segre Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.2 Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.3 The Blow–up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.4 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9 More on Elimination Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.1 The Fundamental Theorem of Elimination Theory . . . . . . . . . . 115
9.2 Morphisms on Projective Varieties Are Closed. . . . . . . . . . . . . 117
9.3 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 118
10 Finite Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
10.1 Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . 119
10.2 Projections and Noether’s Normalization Theorem . . . . . . . . . . 125
10.3 Normal Varieties and Normalization . . . . . . . . . . . . . . . . . . . . 128
10.4 Ramification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
10.5 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Contents ix
11 Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
11.1 Characterization of Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . 137
11.2 Intersection with Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . 138
11.3 Morphisms and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
11.4 Elimination Theory Again. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
11.5 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 151
12 The Cayley Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
12.1 Definition of the Cayley Form. . . . . . . . . . . . . . . . . . . . . . . . . 155
12.2 The Degree of a Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
12.3 The Cayley Form and Equations of a Variety. . . . . . . . . . . . . . 160
12.4 Cycles and Their Cayley Forms. . . . . . . . . . . . . . . . . . . . . . . . 162
12.5 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 164
13 Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
13.1 Plücker Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
13.2 Grassmann Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
13.3 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 182
14 Smooth and Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
14.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
14.2 Some Properties of Smooth Points. . . . . . . . . . . . . . . . . . . . . . 190
14.2.1 Regular Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
14.2.2 System of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 191
14.2.3 Auslander–Buchsbaum Theorem . . . . . . . . . . . . . . . . . 192
14.2.4 Local Equations of a Subvariety . . . . . . . . . . . . . . . . . 195
14.3 Smooth Curves and Finite Maps . . . . . . . . . . . . . . . . . . . . . . . 198
14.4 A Criterion for a Map to Be an Isomorphism. . . . . . . . . . . . . . 200
14.5 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 201
15 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
15.1 Formal Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
15.2 Congruences, Substitution and Derivatives . . . . . . . . . . . . . . . . 207
15.2.1 Conguences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
15.2.2 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
15.2.3 Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
15.3 Fractional Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
15.4 Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 216
16 Affine Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
16.1 Multiple Points and Principal Tangent Lines. . . . . . . . . . . . . . . 219
16.2 Parametrizations and Branches of a Curve . . . . . . . . . . . . . . . . 221
16.3 Intersections of Affine Curves . . . . . . . . . . . . . . . . . . . . . . . . . 224
16.3.1 Intersection Multiplicity and Resultants . . . . . . . . . . . . 225
16.3.2 Order of a Curve at a Branch and Intersection
Multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228