本书是作者在清华大学讲授的研究生课程“代数几何Ⅰ”，的讲义。每次伴随着课程的讲授，作者都要进行修订。经过四五次的修订锤炼之后，作者终于决定出版此书。
交换代数和代数几何是密不可分的，因此阅读本书需要一些交换代数的预备知识。读者通过学习代数几何不仅进一步学习了交换代数。还学习了从几何角度思考交换代数。
作者Looijenga教授是当代最优秀的代数几何学家之一，曾在1978年的国际数学家大会和1992年的欧洲数学家大会做邀请报告。他是荷兰皇家艺术和科学院院士，现任教于清华大学。

1 Affine varieties
1.1 The Zariski topology
1.2 Irreducibility and decomposition
1.3 Finiteness properties and the Hilbert theorems
1.4 The affine category
1.5 The sheaf of regular functions
1.6 The product
1.7 The notion of a variety
1.8 Function fields and rational maps
1.9 Finite morphisms
1.10 Normalization
2 Local properties of varieties
2.1 Dimension
2.2 Smooth and singular points
2.3 Differentials and derivations
2.4 Sheaves of rings and modules
3 Projective varieties
3.1 Projective spaces
3.2 The Zariski topology on a projective space
3.3 The Segre embeddings
3.4 The "proj" construction and blowing up
3.5 Elimination theory and projections
3.6 The Veronese embeddings
3.7 Grassmannians
3.8 Fano varieties and the Gauβ map
3.9 Multiplicities of modules
3.10 Hilbert functions and Hilbert polynomials
4 Projective curves
4.1 Valuations and points
4.2 Divisors and invertible modules on a curve
4.3 The Riemann-Roch theorem
4.4 Residues and Serre duality
4.5 Some applications of the Riemann-Roch theorem .
4.6 The theorem of Riemann-Hurwitz
Index
Bibliography