The Red Book of Varieties and Schemes

The Red Book of Varieties and Schemes presents David Mumford's landmark 1965 Harvard lectures on algebraic geometry. The text develops the foundations of scheme theory and explores the geometry of varieties through concrete examples. The book contains detailed treatments of nilpotents, local rings, and the functorial approach to algebraic geometry. Mumford introduces key concepts through specific calculations rather than abstract theory, making complex ideas accessible through worked examples. The lectures build systematically from basic definitions to advanced topics in modern algebraic geometry. Original exercises and problems appear throughout the text to reinforce the material. The work stands as a bridge between classical and modern approaches to algebraic geometry, demonstrating how abstract machinery connects to geometric intuition. Its influence on the development of scheme theory and algebraic geometry continues to resonate through contemporary mathematics.

- A challenging but rewarding text for advanced algebraic geometry students. Some readers say it revolutionized their understanding of schemes. Likes: - Clear progression from classical varieties to abstract schemes - Concrete examples help bridge the transition - Dense information packed into a concise format - Hand-drawn illustrations aid visualization - Well-structured exercises build comprehension Dislikes: - Too terse for self-study - Requires strong prerequisites in commutative algebra - Some notation and terminology now outdated - Limited discussion of motivations and intuition - Physical copies hard to find and expensive One reader noted: "The exercises do most of the teaching. The text itself is sparse but the problems develop the key ideas." Ratings: Goodreads: 4.4/5 (32 ratings) Amazon: 4.5/5 (6 ratings) Mathematics Stack Exchange: Frequently recommended for scheme theory, but with cautions about difficulty level Most readers suggest pairing it with more expansive modern texts like Hartshorne or Vakil.

Algebraic Geometry by Robin Hartshorne A foundational text that develops the theory of schemes with the same geometric intuition and rigor found in Mumford's Red Book.

Lectures on Curves on an Algebraic Surface by David Mumford This monograph extends the geometric concepts from the Red Book to study curves on surfaces using intersection theory and cohomology.

The Geometry of Schemes by David Eisenbud and Joe Harris The text provides concrete examples and geometric interpretations of scheme theory that complement the theoretical foundations in the Red Book.

Introduction to Commutative Algebra by Michael Atiyah and Ian MacDonald This work presents the algebraic prerequisites for scheme theory with a focus on the connections to geometric concepts.

Fundamental Algebraic Geometry: Grothendieck's FGA Explained by Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli The book expands on the foundations laid in the Red Book by exploring Grothendieck's approach to moduli spaces and relative schemes.

🔸 The Red Book of Varieties and Schemes, published in 1999, began as David Mumford's lecture notes from Harvard in 1961-1962, circulating informally for decades before official publication. 🔸 David Mumford won the Fields Medal in 1974, mathematics' highest honor, for his groundbreaking work in algebraic geometry—the very subject this book helps explain. 🔸 The text introduced many mathematicians to the revolutionary language of schemes, developed by Alexander Grothendieck, which transformed modern algebraic geometry. 🔸 The book's distinctive red cover and informal style made it a beloved classic among graduate students, earning it the nickname "The Red Book" years before its official publication. 🔸 Though written at a graduate level, the book is famous for its clear explanations and carefully chosen examples, making complex concepts in scheme theory more accessible to students.