The Rising Sea: Foundations of Algebraic Geometry

by Ravi Vakil

Overview

An accessible, motivated introduction to one of the most dynamic areas of mathematics

Decades ago, Mumford wrote that algebraic geometry “seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics.” The revolution has now fully come to pass and has fundamentally changed how we think about many fields of mathematics. This book provides a thorough foundation in the powerful ideas that now shape the landscape, with an informal yet rigorous exposition that builds intuition for understanding the formidable machinery. It begins with a discussion of categorical thinking and sheaves and then develops the notion of schemes and varieties as examples of “geometric spaces” before discussing their specific aspects. The book goes on to cover topics such as dimension and smoothness, vector bundles and their natural generalizations, and important cohomological tools and their applications. Important optional topics are included in starred sections.

• Provides a comprehensive introduction certain to become the standard on the subject
• Features a wealth of exercises that enable students to learn by doing
• Requires few prerequisites, developing the tools students need to succeed, from category theory and sheaf theory to commutative and homological algebra
• Uses an example-driven approach that builds mathematical intuition
• Is a self-contained textbook for graduate students and an essential reference for researchers

Product Details

ISBN-13:	9780691268668
Publisher:	Princeton University Press
Publication date:	10/21/2025
Pages:	688
Product dimensions:	7.00(w) x 10.00(h) x (d)

About the Author

Ravi Vakil is the Robert Grimmett Professor of Mathematics at Stanford University and president of the American Mathematical Society. He is the author of A Mathematical Mosaic: Patterns and Problem Solving.

What People are Saying About This

From the Publisher

“Vakil’s book is a thorough introduction to the foundations of modern algebraic geometry as conceived by Grothendieck, emphasizing the intuition of a scheme as a functor of points. It is written for the sophisticated learner, with plenty of advice, examples, wisdom, and exercises. My students love this book! I wish it were available when I was learning the subject.”—Karen E. Smith, coauthor of Rational and Nearly Rational Varieties

“Algebraic geometry has no shortage of excellent textbooks. But even among these, Vakil’s book distinguishes itself by its clear and thorough explanations and its modern coverage of important topics. Vakil, a leader in the field and a master of exposition, imparts insights valuable to both the beginner and the expert. Student-tested over fifteen years, the book is an excellent choice for the classroom or self-study. I expect to continue using it in my own classes.”—Bjorn Poonen, Massachusetts Institute of Technology

“When I was starting out, algebraic geometry was clearly a rich area of study but also hard to learn. Fortunately, the number of people who want to learn the subject continues to increase, and Vakil’s book smooths and speeds the way in. It speaks the modern language of schemes and is generous with proofs and explanations, from category theory to the geometry of curves and surfaces. I recommend the book whenever I teach algebraic geometry.”—Burt Totaro, University of California, Los Angeles

“In The Rising Sea, Ravi Vakil expertly guides readers through the intricate terrain of modern algebraic geometry with remarkable clarity and insight. With engaging commentary and thoughtfully designed exercises, Vakil makes complex ideas accessible, providing intuitive explanations rooted in geometric intuition. The book will serve as a valuable companion for anyone exploring the vast realm of algebraic geometry.”—June Huh, Princeton University