Introductory textbooks on algebraic geometry typically demand a strong mathematical background and can be challenging even for advanced students. While many excellent texts aim to bridge the gap to mastering this rich field, learners who are new to abstract algebra or who have never studied it through a geometric lens still often find the subject inaccessible. Beginning in Algebraic Geometry achieves a remarkable balance, offering a rigorous and detailed development of algebraic geometry that is nevertheless intended to be readable by students with only a first course in abstract algebra and linear algebra as prerequisites. Starting from the most fundamental properties of polynomials, the reader is guided one step at a time through affine, projective, and quasiprojective algebraic geometry, with complete justifications along the way of such foundational results as the Nullstellensatz and the Theorem on Fiber Dimensions.
Several features of this text ensure that it is accessible to the widest possible audience. First, the electronic edition is freely available through Open Access. Furthermore, the authors have skillfully crafted a narrative-driven exposition that reinforces key algebraic concepts (such as quotient rings and modules) and introduces others (such as tensor products and integrality) by developing them within a geometric framework. Well-integrated examples and beautiful illustrations enhance the learning experience, and the writing balances rigor and intuition to maximize readability. Each chapter begins with clearly-stated learning objectives, providing students with a roadmap, and key definitions and results are highlighted for ease of reference. The exercises range from basic to intermediate in difficulty, ensuring sufficient practice without overwhelming the learner. This textbook is suitable for both classroom instruction and independent learners, and it serves as an excellent entry point into the more advanced texts on algebraic geometry.
0 Polynomial rings.- I. Affine Algebraic Geometry.- 1. Varieties and ideals.- 2 Irreducibility of affine varieties.- 3 Coordinate rings.- 4 Polynomial maps.- 5. Proof of the Nullstellensatz.- 6 Dimension.- 7 Smoothness.- 8 Products.- II Projective Algebraic Geometry.- 9 Projective varieties.- 10 Maps of projective varieties.- 11 Quasiprojective varieties.- 12 Culminating topics.- Coda: Where to go from here?.- Index.- Bibliography.
Emily Clader is Associate Professor of Mathematics at San Francisco State University.
Dustin Ross is Associate Professor of Mathematics at San Francisco State University.
This is a wonderful undergraduate level textbook which provides an accessible introduction to algebraic geometry. ... The text contains a good supply of exercises which supplements the narrative ... . The text under review thus provides a fresh point of entry to algebraic geometry. It is an important contribution to the existing textbook literature and is suitable for use as an adopted text in a course and/or for use as a directed reading and/or as a self-study text. (Nathan Grieve, Mathematical Reviews, April, 2026)
