This book consists of two parts. The first is devoted to an introduction to basic concepts in algebraic geometry: affine and projective varieties, some of their main attributes and examples. The second part is devoted to the theory of curves: local properties, affine and projective plane curves, resolution of singularities, linear equivalence of divisors and linear series, Riemann–Roch and Riemann–Hurwitz Theorems.
The approach in this book is purely algebraic. The main tool is commutative algebra, from which the needed results are recalled, in most cases with proofs. The prerequisites consist of the knowledge of basics in affine and projective geometry, basic algebraic concepts regarding rings, modules, fields, linear algebra, basic notions in the theory of categories, and some elementary point–set topology.
This book can be used as a textbook for an undergraduate course in algebraic geometry. The users of the book are not necessarily intended to become algebraic geometers but may be interested students or researchers who want to have a first smattering in the topic.
The book contains several exercises, in which there are more examples and parts of the theory that are not fully developed in the text. Of some exercises, there are solutions at the end of each chapter.
Author(s): Ciro Ciliberto
Series: Unitext 129
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 327
Tags: Introduction, Algebraic Geometry
Preface
Contents
1 Affine and Projective Algebraic Sets
1.1 Affine Algebraic Sets
1.2 Projective Spaces
1.3 Graded Rings
1.4 Projective Algebraic Sets
1.5 Projective Closure of Affine Sets
1.6 Examples
1.6.1 Points
1.6.2 Projective Subspaces
1.6.3 Affine Subspaces
1.6.4 Hypersurfaces
1.6.5 Divisors
1.6.6 Product Topology
1.7 Solutions of Some Exercises
2 Basic Notions of Elimination Theory and Applications
2.1 The Resultant of Two Polynomials
2.2 The Intersection of Two Plane Curves
2.3 Kronecker Elimination Method: One Variable
2.4 Kronecker Elimination Method: More Variables
2.5 Hilbert Nullstellensatz
2.6 Solutions of Some Exercises
3 Zariski Closed Subsets and Ideals in the Polynomials Ring
3.1 Ideals and Coordinate Rings
3.2 Examples
3.2.1 Maximal Ideals
3.2.2 The Twisted Cubic
3.2.3 Cones
3.3 Solutions of Some Exercises
4 Some Topological Properties
4.1 Irreducible Sets
4.2 Noetherian Spaces
4.3 Topological Dimension
4.4 Solutions of Some Exercises
5 Regular and Rational Functions
5.1 Regular Functions
5.2 Rational Functions
5.3 Local Rings
5.4 Integral Elements over a Ring
5.5 Subvarieties and Their Local Rings
5.6 Product of Affine Varieties
5.7 Solutions of Some Exercises
6 Morphisms
6.1 The Definition of Morphism
6.2 Which Maps Are Morphisms
6.3 Affine Varieties
6.4 The Veronese Morphism
6.5 Solutions of Some Exercises
7 Rational Maps
7.1 Definition of Rational Maps and Basic Properties
7.2 Birational Models of Quasi-projective Varieties
7.3 Unirational and Rational Varieties
7.4 Solutions of Some Exercises
8 Product of Varieties
8.1 Segre Varieties
8.2 Products
8.3 The Blow–up
8.4 Solutions of Some Exercises
9 More on Elimination Theory
9.1 The Fundamental Theorem of Elimination Theory
9.2 Morphisms on Projective Varieties Are Closed
9.3 Solutions of Some Exercises
10 Finite Morphisms
10.1 Definitions and Basic Results
10.2 Projections and Noether's Normalization Theorem
10.3 Normal Varieties and Normalization
10.4 Ramification
10.5 Solutions of Some Exercises
11 Dimension
11.1 Characterization of Hypersurfaces
11.2 Intersection with Hypersurfaces
11.3 Morphisms and Dimension
11.4 Elimination Theory Again
11.5 Solutions of Some Exercises
12 The Cayley Form
12.1 Definition of the Cayley Form
12.2 The Degree of a Variety
12.3 The Cayley Form and Equations of a Variety
12.4 Cycles and Their Cayley Forms
12.5 Solutions of Some Exercises
13 Grassmannians
13.1 Plücker Coordinates
13.2 Grassmann Varieties
13.3 Solutions of Some Exercises
14 Smooth and Singular Points
14.1 Basic Definitions
14.2 Some Properties of Smooth Points
14.2.1 Regular Rings
14.2.2 System of Parameters
14.2.3 Auslander–Buchsbaum Theorem
14.2.4 Local Equations of a Subvariety
14.3 Smooth Curves and Finite Maps
14.4 A Criterion for a Map to Be an Isomorphism
14.5 Solutions of Some Exercises
15 Power Series
15.1 Formal Power Series
15.2 Congruences, Substitution and Derivatives
15.2.1 Conguences
15.2.2 Substitution
15.2.3 Derivatives
15.3 Fractional Power Series
15.4 Solutions of Some Exercises
16 Affine Plane Curves
16.1 Multiple Points and Principal Tangent Lines
16.2 Parametrizations and Branches of a Curve
16.3 Intersections of Affine Curves
16.3.1 Intersection Multiplicity and Resultants
16.3.2 Order of a Curve at a Branch and Intersection Multiplicities
16.3.3 More Properties of Branches and of Intersection Multiplicity
16.3.4 Further Interpretation of the Intersection Multiplicity
16.4 Solutions of Some Exercises
17 Projective Plane Curves
17.1 Some Generalities
17.1.1 Recalling Some Basic Definitions
17.1.2 The Bézout Theorem
17.1.3 Linear Systems
17.2 M. Noether's Af+Bg Theorem
17.3 Applications of the Af+Bg Theorem
17.3.1 Pascal's and Pappo's Theorems
17.3.2 The Group Law on a Smooth Cubic
17.4 Solutions of Some Exercises
18 Resolution of Singularities of Curves
18.1 The Case of Ordinary Singularities
18.2 Reduction to Ordinary Singularities
18.2.1 Statement of the Main Theorem
18.2.2 Standard Quadratic Transformations
18.2.3 Transformation of a Curve via a Standard Quadratic Transformation
18.2.4 Proof of the Main Theorem
19 Divisors, Linear Equivalence, Linear Series
19.1 Divisors
19.2 Linear Equivalence
19.3 Fibres of a Morphism
19.4 Linear Series
19.5 Linear Series and Projective Morphisms
19.6 Adjoint Curves
19.7 Linear Systems of Plane Curves and Linear Series
19.8 Solutions of Some Exercises
20 The Riemann–Roch Theorem
20.1 The Riemann–Roch Theorem
20.2 Consequences of the Riemann–Roch Theorem
20.3 Differentials
20.3.1 Algebraic Background
20.3.2 Differentials and Canonical Divisors
20.3.3 The Riemann–Hurwitz Theorem
20.4 Solutions of Some Exercises
Appendix References
Index
