Basic Algebraic Geometry, volume I, is a revised and expanded new edition of the first four chapters of Shafarevich's well-known introductory book on algebraic geometry. Besides correcting misprints and inaccuracies the author has added plenty of new material. Shafarevich succeeds in making algebraic geometry accessible to non-specialists and beginners and his two-volume book will remain one of the most popular introductions to this field. The book is suitable for third-year undergraduates in mathematics and also for students of theoretical physics who would like to learn algebraic geometry.

This book is a revised and expanded new edition of the first four chapters of Shafarevich's well-known introductory book on algebraic geometry. Besides correcting misprints and inaccuracies, the author has added plenty of new material, mostly concrete geometrical material such as Grassmannian varieties, plane cubic curves, the cubic surface, degenerations of quadrics and elliptic curves, the Bertini theorems, and normal surface singularities.

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 - 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 - 3. Rational Functions.- 3.1. Irreducible Algebraic Subsets.- 3.2. Rational Functions.- 3.3. Rational Maps.- Exercises to - 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 - 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 - 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 - 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 - 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 - 3. Properties of Nonsingular Points.- 3.1. Codimension 1 Subvarieties.- 3.2. Nonsingular Subvarieties.- Exercises to - 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 - 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 - 6. Singularities of a Map.- 6.1. Irreducibility.- 6.2. Nonsingularity.- 6.3. Ramification.- 6.4. Examples.- Exercises to - 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 - 2. Divisors on Curves.- 2.1. The Degree of a Divisor on a Curve.- 2.2. Bout's Theorem on a Curve.- 2.3. The Dimension of a Divisor.- Exercises to - 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 - 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 - 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 - 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 - 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 - 2. Applications of Intersection Numbers.- 2.1. Bout'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 - 3. Birational Maps of Surfaces.- 3.1. Blowups of Surfaces.- 3.2. Some Intersection Numbers.- 3.3. Resolution of Indeterminacy.- 3.4. Factorisation as a Chain of Blowups.- 3.5. Remarks and Examples.- Exercises to - 4. Singularities.- 4.1. Singular Points of a Curve.- 4.2. Surface Singularities.- 4.3. Du Val Singularities.- 4.4. Degeneration of Curves.- Exercises to - Algebraic Appendx.- References.