"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 "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.