Igor R. Shafarevich's "Basic Algebraic Geometry 1" is a second, revised and expanded edition of his original Russian work, published by Springer-Verlag. The book provides a comprehensive overview of algebraic geometry, covering both foundational concepts and more specialized topics. It is structured into two volumes, with the first volume focusing on varieties in projective space and the second on schemes and varieties. The book includes a variety of additional material, such as discussions on moduli spaces, Hilbert polynomials, and the Hilbert scheme, as well as a detailed treatment of singularities and their implications. The first volume is divided into four chapters, each exploring different aspects of algebraic geometry. It begins with basic notions, including algebraic curves in the plane, closed subsets of affine space, rational functions, and quasiprojective varieties. It then moves on to products and maps of quasiprojective varieties, dimension, local properties, and the structure of birational maps. The chapter on divisors and differential forms introduces key concepts such as divisors, differential forms, and their applications, including the Riemann-Roch theorem for curves. The second volume covers schemes and varieties, including the definition and properties of schemes, abstract and quasiprojective varieties, coherent sheaves, and the classification of geometric objects. It also discusses the Hilbert polynomial, flat families, and the Hilbert scheme, which are essential tools in algebraic geometry. The book is written with a clear and concise style, making it accessible to readers with a background in undergraduate algebra. It includes a variety of exercises and examples, and the author emphasizes the importance of both geometric intuition and algebraic rigor. The book also includes a historical sketch of the development of algebraic geometry, highlighting key contributions from various mathematicians and the evolution of the field. The translator's note emphasizes the book's role as a foundational text in algebraic geometry, suitable for both students and researchers. It is noted that while the book does not cover every aspect of algebraic geometry in maximal generality, it provides a well-considered selection of topics with a human-oriented discussion of the motivation and ideas. The book is recommended for those seeking a broad understanding of algebraic geometry, as well as for those who need a liberal education in the subject.Igor R. Shafarevich's "Basic Algebraic Geometry 1" is a second, revised and expanded edition of his original Russian work, published by Springer-Verlag. The book provides a comprehensive overview of algebraic geometry, covering both foundational concepts and more specialized topics. It is structured into two volumes, with the first volume focusing on varieties in projective space and the second on schemes and varieties. The book includes a variety of additional material, such as discussions on moduli spaces, Hilbert polynomials, and the Hilbert scheme, as well as a detailed treatment of singularities and their implications. The first volume is divided into four chapters, each exploring different aspects of algebraic geometry. It begins with basic notions, including algebraic curves in the plane, closed subsets of affine space, rational functions, and quasiprojective varieties. It then moves on to products and maps of quasiprojective varieties, dimension, local properties, and the structure of birational maps. The chapter on divisors and differential forms introduces key concepts such as divisors, differential forms, and their applications, including the Riemann-Roch theorem for curves. The second volume covers schemes and varieties, including the definition and properties of schemes, abstract and quasiprojective varieties, coherent sheaves, and the classification of geometric objects. It also discusses the Hilbert polynomial, flat families, and the Hilbert scheme, which are essential tools in algebraic geometry. The book is written with a clear and concise style, making it accessible to readers with a background in undergraduate algebra. It includes a variety of exercises and examples, and the author emphasizes the importance of both geometric intuition and algebraic rigor. The book also includes a historical sketch of the development of algebraic geometry, highlighting key contributions from various mathematicians and the evolution of the field. The translator's note emphasizes the book's role as a foundational text in algebraic geometry, suitable for both students and researchers. It is noted that while the book does not cover every aspect of algebraic geometry in maximal generality, it provides a well-considered selection of topics with a human-oriented discussion of the motivation and ideas. The book is recommended for those seeking a broad understanding of algebraic geometry, as well as for those who need a liberal education in the subject.