Bibliography
- Commutative algebra
- Atiyah and Macdonald: Introduction to Commutative Algebra
(It is an excellent textbook. Moreover, It is fair to say that one needs at least what is covered in this textbook to get into most areas of algebraic geometry.)
- David Eisenbud Commutative Algebra with a View Toward Algebraic Geometry
(Beautiful book, which form a good combination with the previous one to learn commutative algebra)
- Allen ALTMAN and Steven KLEIMANA: Term of Commutative Algebra (There is a book on Commutative Algebra written in the style of Atiyah/MacDonald but with more categorical language by two professors at MIT. It is written very concisely with full solutions to the exercises -- this might be a good reference for solved examples: http://web.mit.edu/18.705/www/syl13f.html. The first half covers some ring, categories and limits, localization.)
- Kemper: A Course in Commutative Algebra
(its originally freely downloadable draft version included lots of exercises and problems with all their solutions (the published version does not).)
- Singh: Basic Commutative Algebra.
(A nice new succinct textbook to use as a purely formal reference (but with doable exercises).)
- Reid: Undergraduate Commutative Algebra.
(A good geometry-motivated introduction)
- Category theory
- Ravi Vakil: Foundations of algebraic geometry (Chapter 1, good introduction in category theory)
- Pierre Schapira: Categories and Homological Algebra (Complete reference to learn about categories)
- Algebraic geometry
- Beginners
- Perrin: Algebraic Geometry
(It has good examples and lots of (doable) exercises (like some very interesting and easy motivations for cohomology to solve conceptually geometric problems). This is a more algebraic treatment but which develops almost all needed concepts along the way and it is one of the best introductions to the language of sheaf theory in algebraic geometry without the need of schemes (but even defines finite schemes to define multiplicities, letting you understand the relationship of how Beltrametti does things and why jumping into schemes is a good idea in the end). So eventually, you end up understanding degrees, genus, Bézout and Riemann-Roch among other things through exact sequences)
- Harris: Algebraic Geometry, A First Course
(one cannot learn much from this book alone if it is not supplemented by standard theoretical titles like the ones above. Nevertheless it is a great source for classical examples and results which does a good job at complementing the other books as a companion.)
- Miles Reid: Undergraduate algebraic geometry
(really good introduction to the subject)
- Joe Harris: Algebraic Geometry: A First Course
(This book, however, emphasizes the classical roots of the subject but if you have not yet seen too much of algebraic geometry, it is worthwhile getting this book and reading a few lectures. (The book is split into "lectures" rather than "chapters".) There are many beautiful constructions in classical algebraic geometry that can be understood without too much background (and which lay the foundations for some aspects of modern algebraic geometry) and this can perhaps give you a rough indication of the geometric intuitions in algebraic geometry. The book does an excellent job of conveying the beauty and elegance of algebraic geometry.)
- Curves and surfaces
- Kirwan: Complex Algebraic Curves.
(This is an introduction to algebraic geometry through Riemann surfaces/complex algebraic curves)
- Beltrametti et al.: Lectures on Curves, Surfaces and Projective Varieties
(It develops CLASSICAL algebraic geometry using a minimal amount of commutative algebra (you barely need some notions of abstract algebra). It is developed in the most similar way to how the subject was done before the Grothendieck revolution which seems to have permeated every modern title. There you can find old definitions of genus as an "excess", a nonstandard (noncohomological) treatment of Riemann-Roch for curves, classical important examples and constructions for curves, surfaces and morphisms, and even the development of multiplicities and Bézout's theorem using old good elimination theory and proving its projective invariance arriving at the formula most modern books use to define multiplicity (and thus forgetting all classical motivation whatsoever, the dimension of a particular vector space).)
- Fulton's Algebraic Curves: An Introduction to Algebraic Geometry
(Freely available)
- Advance, classical references
- Hartshorne: Algebraic Geometry
- Qing Liu: Algebraic Geometry and Arithmetic Curves
- Grothendieck: EGA, SGA, FGA