Basic Algebraic Geometry 1

by Igor R. Shafarevich

I. Basic Notions.- 1. Algebraic Curves in the Plane.- 1.1. Plane Curves.- 1.2. Rational Curves.- 1.3. Relation with Field Theory.- 1.4. Rational Maps.- 1.5. Singular and Nonsingular Points.- 1.6. The Projective Plane.- Exercises to §1.- 2. Closed Subsets of Affine Space.- 2.1. Definition of Closed Subsets.- 2.2. Regular Functions on a Closed Subset.- 2.3. Regular Maps.- Exercises to §2.- 3. Rational Functions.- 3.1. Irreducible Algebraic Subsets.- 3.2. Rational Functions.- 3.3. Rational Maps.- Exercises to §3.- 4. Quasiprojective Varieties.- 4.1. Closed Subsets of Projective Space.- 4.2. Regular Functions.- 4.3. Rational Functions.- 4.4. Examples of Regular Maps.- Exercises to §4.- 5. Products and Maps of Quasiprojective Varieties.- 5.1. Products.- 5.2. The Image of a Projective Variety is Closed.- 5.3. Finite Maps.- 5.4. Noether Normalisation.- Exercises to §5.- 6. Dimension.- 6.1. Definition of Dimension.- 6.2. Dimension of Intersection with a Hypersurface.- 6.3. The Theorem on the Dimension of Fibres.- 6.4. Lines on Surfaces.- Exercises to §6.- II. Local Properties.- 1. Singular and Nonsingular Points.- 1.1. The Local Ring of a Point.- 1.2. The Tangent Space.- 1.3. Intrinsic Nature of the Tangent Space.- 1.4. Singular Points.- 1.5. The Tangent Cone.- Exercises to §1.- 2. Power Series Expansions.- 2.1. Local Parameters at a Point.- 2.2. Power Series Expansions.- 2.3. Varieties over the Reals and the Complexes.- Exercises to §2.- 3. Properties of Nonsingular Points.- 3.1. Codimension 1 Subvarieties.- 3.2. Nonsingular Subvarieties.- Exercises to §3.- 4. The Structure of Birational Maps.- 4.1. Blowup in Projective Space.- 4.2. Local Blowup.- 4.3. Behaviour of a Subvariety under a Blowup.- 4.4. Exceptional Subvarieties.- 4.5. Isomorphism and Birational Equivalence.- Exercises to §4.- 5. Normal Varieties.- 5.1. Normal Varieties.- 5.2. Normalisation of an Affine Variety.- 5.3. Normalisation of a Curve.- 5.4. Projective Embedding of Nonsingular Varieties.- Exercises to §5.- 6. Singularities of a Map.- 6.1. Irreducibility.- 6.2. Nonsingularity.- 6.3. Ramification.- 6.4. Examples.- Exercises to §6.- III. Divisors and Differential Forms.- 1. Divisors.- 1.1. The Divisor of a Function.- 1.2. Locally Principal Divisors.- 1.3. Moving the Support of a Divisor away from a Point ....- 1.4. Divisors and Rational Maps.- 1.5. The Linear System of a Divisor.- 1.6. Pencil of Conics over ?1.- Exercises to §1.- 2. Divisors on Curves.- 2.1. The Degree of a Divisor on a Curve.- 2.2. Bézout's Theorem on a Curve.- 2.3. The Dimension of a Divisor.- Exercises to §2.- 3. The Plane Cubic.- 3.1. The Class Group.- 3.2. The Group Law.- 3.3. Maps.- 3.4. Applications.- 3.5. Algebraically Nonclosed Field.- Exercises to §3.- 4. Algebraic Groups.- 4.1. Algebraic Groups.- 4.2. Quotient Groups and Chevalley's Theorem.- 4.3. Abelian Varieties.- 4.4. The Picard Variety.- Exercises to §4.- 5. Differential Forms.- 5.1. Regular Differential 1-forms.- 5.2. Algebraic Definition of the Module of Differentials.- 5.3. Differential p-forms.- 5.4. Rational Differential Forms.- Exercises to §5.- 6. Examples and Applications of Differential Forms.- 6.1. Behaviour Under Maps.- 6.2. Invariant Differential Forms on a Group.- 6.3. The Canonical Class.- 6.4. Hypersurfaces.- 6.5. Hyperelliptic Curves.- 6.6. The Riemann-Roch Theorem for Curves.- 6.7. Projective Embedding of a Surface.- Exercises to §6.- IV. Intersection Numbers.- 1. Definition and Basic Properties.- 1.1. Definition of Intersection Number.- 1.2. Additivity.- 1.3. Invariance Under Linear Equivalence.- 1.4. The General Definition of Intersection Number.- Exercises to §1.- 2. Applications of Intersection Numbers.- 2.1. Bézout's Theorem in Projective and Multiprojective Space.- 2.2. Varieties over the Reals.- 2.3. The Genus of a Nonsingular Curve on a Surface.- 2.4. The Riemann-Roch Inequality on a Surface.- 2.5. The Nonsingular Cubic Surface.- 2.6. The Ring of Cycle Classes.- Exercises to §2.- 3. Bi

Genres: Mathematics, Science, Textbooks

324 pages, Paperback

First published August 8, 1994

About the author

Igor Rostislavovich Shafarevich is a Russian mathematician who has contributed to algebraic number theory and algebraic geometry. He has written books and articles that criticize socialism and was an important dissident during the Soviet regime.

Reviews

[George · Rating 4 out of 5]

This is a great book to learn from in a second course on varieties, or perhaps, after skimming a 'lighter' alternative. It is quite heavy and often a bit hard to follow. However, it is very complete, appeals to intuition, and often prefers taking the geometrical side over the algebraic, making it certainly easier to get a 'big picture', at the expense of making the proofs more complicated. Certainly I'm not the smartest person, but I often felt that I would not have been able to produce most of the proofs of the theorems, and after a thorough reading I was still left slightly uneasy. This is not a jab at the translators, but at times it is quite obvious that some details are lost in translation and the grammar to make it work seems off, which further exacerbated the issue.

I admit that I had some trouble with the often-changing non-standard notation presented in the textbook. For example, I had a hard time following his examples of the Grassmannian and Veronese varieties since the notation was so clunky and unexplained. The exercises tend to be either easy or hard, but at least you can count on there being plenty of them.

This was my experience with the first two chapters - I haven't read the last two, and it was recommended to me to not do so by a well-known geometer if one expects to go further in their studies of AG. It is better to learn from a more general source such as Hartshorne in that case, or other, more modern books on the subject.