The book "Basic Algebraic Geometry 1" by Igor R. Shafarevich is a comprehensive introduction to the foundations of algebraic geometry. The second, revised, and expanded edition, translated by Miles Reid, covers the essential aspects of the subject, providing a broad overview without delving too deeply into specialized theories. The book is divided into three parts, with the first part focusing on varieties in projective space, the second part on schemes and varieties, and the third part on complex algebraic varieties and complex manifolds. Key topics include: - Basic notions of algebraic curves and varieties in projective space. - Local properties of varieties, including singular and nonsingular points, birational maps, and normal varieties. - Divisors and differential forms, with applications to curves and surfaces. - Intersection numbers and their applications. - Schemes and their properties, including sheaves and coherent sheaves. - Complex topology of algebraic varieties and complex manifolds, including connectedness, homology, and uniformization. The book aims to provide a well-rounded introduction to the subject, suitable for both undergraduate and graduate students, and includes numerous exercises and examples to illustrate the concepts. It also includes an algebraic appendix and a historical sketch of the development of algebraic geometry.The book "Basic Algebraic Geometry 1" by Igor R. Shafarevich is a comprehensive introduction to the foundations of algebraic geometry. The second, revised, and expanded edition, translated by Miles Reid, covers the essential aspects of the subject, providing a broad overview without delving too deeply into specialized theories. The book is divided into three parts, with the first part focusing on varieties in projective space, the second part on schemes and varieties, and the third part on complex algebraic varieties and complex manifolds. Key topics include: - Basic notions of algebraic curves and varieties in projective space. - Local properties of varieties, including singular and nonsingular points, birational maps, and normal varieties. - Divisors and differential forms, with applications to curves and surfaces. - Intersection numbers and their applications. - Schemes and their properties, including sheaves and coherent sheaves. - Complex topology of algebraic varieties and complex manifolds, including connectedness, homology, and uniformization. The book aims to provide a well-rounded introduction to the subject, suitable for both undergraduate and graduate students, and includes numerous exercises and examples to illustrate the concepts. It also includes an algebraic appendix and a historical sketch of the development of algebraic geometry.