Algebraic Geometry is a branch of mathematics which studies algebraic varieties, which are graphs of solutions to sets of multivariate polynomial equations. The roots of this field go back to ancient greece, when Menaechmus solved the problem of finding a cube of side x whose volume is equal to a rectangle of sides a,a, and b by intersecting the parabola ay=x^2 and the hyperbola xy=ab. Sharaf al-Din was a scholar during the Islamic Golden Age who solved a cubic f(x)=c by locating a maximum point m and showing there were two solutions if f(m) > c, one if f(m)=c, and none if f(m) < c. Relying heavily on commutative algebra in the 20th century, methods of algebraic geometry were developed by Zariski and Weil, and later revolutionized with the scheme theory from Grothendieck and Serre. These advances were critical in proving finally Fermat's Last Theorem using techniques cross-developed with algebraic number theory.
Prerequisites: Algebraic Geometry is an advanced subject to learn and no single book is going to give you everything you need. Some of the commutative algebra is taught as needed, but still a strong background in abstract algebra is required.
- Cox, Littlewood, and O'Shea, "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
- Shafarevich and Reid, Basic Algebraic Geometry 1: Varieties in Projective Space
- Harris, Algebraic Geometry: A First Course (full-text)
- Griffiths and Harris, Principles of Algebraic Geometry
- Mumford, Algebraic Geometry I: Complex Projective Varieties
- Cox, Little, and Schenck, Toric Varieties
- Eisenbud and Harris, Geometry of Schemes
- Hartshorne, Algebraic Geometry (very difficult)
- Dr. Balaji, What is Algebraic Geometry (video lecture series)
- Wikipedia, Italian School of Algebraic Geometry (good bibliography)
- Dieudonne, The Historical Development of Algebraic Geometry
- Grothendieck, Éléments de géométrie algébrique (EGA vol. 1, french, very difficult)
(Note: All volumes of EGA are available for free from the above link in sequence. Djvu viewer required.)