本书是沙法列维奇代数几何基础教程的第1卷。本书作者沙法列维奇是当代著名数学家，被誉为苏联数学三巨头之一，他建立了为世人瞩目的苏联代数几何和代数数论学派。他的代数几何基础教程（俄文版1972年初版，英文版1977年初版）问世五十多年来，历经三版修订，一直被视为一部重要的代数几何经典名著。与同类教材相比，该教程内容全面详尽，注重给出抽象理论的几何背景和起源，并配有充分反映几何本质的实例和图解。本书所需预备知识仅限于代数基础，是高年级本科生和研究生学习代数几何的首选教材。该教程的第3版分为两卷，第1卷讨论射影空间中的簇，第2卷讨论概形和复流形。

Book 1: Varieties in Projective Space
1 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
1.7 Exercises to Section 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
2.4 Exercises to Section 2
3 Rational Functions
3.1 Irreducible Algebraic Subsets
3.2 Rational Functions
3.3 Rational Maps
3.4 Exercises to Section 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
4.5 Exercises to Section 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
5.5 Exercises to Section 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
6.5 Exercises to Section 6
2 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
1.6 Exercises to Section 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
2.4 Exercises to Section 2
3 Properties of Nonsingular Points
3.1 Codimension 1 Subvarieties
3.2 Nonsingular Subvarieties
3.3 Exercises to Section 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
4.6 Exercises to Section 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
5.5 Exercises to Section 5
6 Singularities of a Map
6.1 Irreducibility
6.2 Nonsingularity
6.3 Ramification
6.4 Examples
6.5 Exercises to Section 6
……
Book 2: Schemes and Varieties
Book 3: Complex Algebraic Varieties and Complex Manifolds