Introduction to Intersection Theory in Algebraic Geometry

Introduction to Intersection Theory in Algebraic Geometry is a mathematics textbook that presents the foundations and key concepts of intersection theory. The work outlines methods for calculating intersections of algebraic varieties and subvarieties in projective space. The book progresses from basic definitions through increasingly complex mathematical concepts, including Chow groups, proper pushforward, and intersection products. The text incorporates numerous examples and exercises to reinforce the theoretical material. Each chapter builds systematically on previous material while maintaining clear connections between different aspects of intersection theory. The author includes historical notes and references to related mathematical developments. This text serves as a bridge between classical algebraic geometry and modern intersection theory, establishing essential tools for understanding more advanced concepts in algebraic geometry. The work's approach emphasizes both rigor and accessibility for graduate-level mathematics students.

Readers describe this as a dense, technical text that requires significant mathematical maturity and background knowledge in algebraic geometry. One doctoral student noted it was "impenetrable" without first studying easier intersection theory texts. Likes: - Clear progression of concepts - Comprehensive coverage of classical and modern methods - Useful exercises throughout - High-quality typesetting and layout Dislikes: - Assumes too much prior knowledge - Little motivation for concepts - Few worked examples - Some proofs lack detail Ratings: Goodreads: 4.0/5 (12 ratings) Amazon: Not enough reviews for rating Multiple reviewers recommend reading Eisenbud & Harris's "3264 & All That" first for a gentler introduction. A mathematics professor on Mathematics Stack Exchange called it "the standard reference but not the best first book on the subject." Mathematical Reviews praised its thoroughness but noted its demanding nature.

Intersection Theory by Helmut Hironaka A systematic development of intersection theory focusing on singular varieties and resolution of singularities.

3264 and All That: A Second Course in Algebraic Geometry by David Eisenbud and Joe Harris The text explores intersection theory through enumerative geometry problems and Schubert calculus applications.

Principles of Algebraic Geometry by Phillip Griffiths and Joseph Harris An introduction to complex algebraic geometry that connects intersection theory with complex manifolds and differential geometry.

Intersection Theory in Algebraic Stacks by Roy Joshua A treatment of intersection theory in the modern context of algebraic stacks and derived categories.

Intersection Cohomology by Armand Borel The book presents intersection homology theory and its connections to intersection theory and perverse sheaves.

🌟 William Fulton wrote this influential text while at Brown University, where he helped establish one of the strongest algebraic geometry programs in the United States. 🌟 Intersection theory, the book's focus, has deep connections to string theory in physics and has become crucial in modern theoretical physics research. 🌟 The techniques presented in this book helped solve several classical problems, including the computation of numbers that had puzzled mathematicians since the 19th century about how many curves of given degree satisfy various geometric conditions. 🌟 Despite being published in 1984, this text remains one of the most recommended introductory books for graduate students studying algebraic geometry, particularly praised for its clear explanations of complex concepts. 🌟 Fulton later expanded many concepts from this book in his more comprehensive work "Intersection Theory" (1998), which won the Steele Prize for Mathematical Exposition from the American Mathematical Society.