Algebraic Geometry

Below is a list of reviews of books on algebraic geometry.

Ideals, Varieties and Algorithms
Cox, D., Little, J., O’Shea, D. (4th ed. 2015) ISBN 9783319167206
Prerequisites: Linear Algebra and some mathematical maturity, abstract algebra recommended

This is an undergraduate-friendly introduction to commutative algebra and algebraic geometry, notably not even making assumptions about whether the student has had a first algebra course. The book spends a little while building up in particular the theory of Groebner bases, and heavily emphasizes computational methods: The chapter on elimination theory, for instance, gives a specific algorithm in pseudocode for finding the solutions to a system of polynomials. The book takes a while to build up to some more classical ideas such as the Nullstellensatz (which is not covered until over 150 pages in), quotient rings (which are not covered until over 200 pages in) and projective varieties (which it takes almost 400 pages to get to), but that is in large part because it again assumes essentially no background in algebra beyond linear algebra. It also has interesting applications, such as to robotics and to invariant theory, and has many exercises that both reinforce the content of the text and explore further in interesting ways. A one semester course using the book should be able to cover the first four chapters with time leftover, after which there are several directions one could go in; a year-long course should have no difficulty finishing the book, and could benefit from the list of project ideas in the appendix at the end.