Undergraduate texts.
Bix’sConics and Cubics: A Concrete Introductionto Algebraic Geometry[Bix98] concentrates on the zero loci of second degree (conics) and third degree (cubics) two variable polynomials. This is a true undergraduate text. Bix shows the classical fact, as we will see, that smooth conics (i.e., ellipses, hyperbolas, and parabolas) are all equivalent under a projective change of coordinates. He then turns to cubics, which are much more difficult, and shows in particular how the points on a cubic form an abelian group. For even more leisurely introductions to second degree curves, seeAkopyan and Zaslavsky’sGeometry of Conics[AZ07] and Kendig’s Conics [Ken].
Reid’sUndergraduate Algebraic Geometry[Rei88] is another good text, though the undergraduate in the title refers to British undergraduates, who start to concentrate in mathematics at an earlier age than their U.S. counterparts. Reid starts with plane curves, shows why the natural ambient space for these curves is projective space, and then develops some of the basic tools needed for higher dimensional varieties. His brief history of algebraic geometry is also fun to read.
Ideals, Varieties, and Algorithms: An Introduction to ComputationalAlgebraic Geometry and Commutative Algebraby Cox, Little, and O’Shea [CLO07] is almost universally admired. This book is excellent at explaining Groebner bases, which is the main tool for producing algorithms in algebraic geometry and has been a major theme in recent research. It might not be the best place for the rank beginner, who might wonder why these algorithms are necessary and interesting.
An Invitation to Algebraic Geometryby K. Smith, L. Kahanpaa, P. Kekaelaeinen, and W. N. Traves [SKKT00] is a wonderfully intuitive book, stressing the general ideas. It would be a good place to start for any student who has completed a first course in algebra that included ring theory.
Gibson’sElementary Geometry of Algebraic Curves: An UndergraduateIntroduction[Gib98] is also a good place to begin.
There is also Hulek’sElementary Algebraic Geometry[Hul03], though this text might be more appropriate for German undergraduates (for whom it was written) than U.S. undergraduates. The most recent of these books is Hassett’s Introduction to Algebraic Geometry, [Has07], which is a good introductory text for students who have taken an abstract algebra course.
Graduate texts.
There are a number, though the first two on the list have dominated the market for the last 35 years. Hartshorne’sAlgebraic Geometry[Har77] relies on a heavy amount of commutative algebra. Its first chapter is an overview of algebraic geometry, while chapters four and five deal with curves andsurfaces, respectively. It is in chapters two and three that the heavy abstract machinery that makes much of algebraic geometry so intimidating is presented. These chapters are not easy going but vital to get a handle on the Grothendieck revolution in mathematics. This should not be the first source for learning algebraic geometry; it should be the second or third source. Certainly young budding algebraic geometers should spend time doing all of the homework exercises in Hartshorne; this is the profession’s version of paying your dues.
Principles of Algebraic Geometryby Griffiths and Harris [GH94] takes a quite different tack from Hartshorne. The authors concentrate on the several complex variables approach. Chapter zero in fact is an excellent overview of the basic theory of several complex variables. In this book analytic tools are freely used, but an impressive amount of geometric insight is presented throughout.
Shafarevich’sBasic Algebraic Geometryis another standard, longtime favorite, now split into two volumes, [Sha94a] and [Sha94b]. The first volume concentrates on the relatively concrete case of subvarieties in complex projective space, which is the natural ambient space for much of algebraic geometry. Volume II turns to schemes, the key idea introduced by Grothendieck that helped change the very language of algebraic geometry.
Mumford’sAlgebraic Geometry I: Complex Projective Varieties[Mum95] is a good place for a graduate student to get started. One of the strengths of this book is how Mumford will give a number of definitions, one right after another, of the same object, forcing the reader to see the different reasonable ways the same object can be viewed.
Mumford’sThe Red Book of Varieties and Schemes[Mum99]was for many years only available in mimeograph form from Harvard’sMathematics Department, bound in red (hence its title “The Red Book”), though it is now actually yellow. It was prized for its clear explanation of schemes. It is an ideal second or third source for learning about schemes. This new edition includes Mumford’s delightful book Curves and their Jacobians, which is a wonderful place for inspiration.
Fulton’sAlgebraic Curves[Ful69] is a good brief introduction. When it was written in the late 1960s, it was the only reasonable introduction to modern algebraic geometry.
Miranda’sAlgebraic Curves and Riemann Surfaces[Mir95] is a popular book, emphasizing the analytic side of algebraic geometry.
Harris’sAlgebraic Geometry: A First Course[Har95] is chockfull of examples. In a forest versus trees comparison, it is a book of trees. This makes it difficult as a first source, but ideal as a reference for examples.
Ueno’s two volumes,Algebraic Geometry 1: From Algebraic Varietiesto Schemes[Uen99] andAlgebraic Geometry 2: Sheaves andCohomology[Uen01], will lead the reader to the needed machinery for much of modern algebraic geometry.
Bump’sAlgebraic Geometry[Bum98], Fischer’sPlane AlgebraicCurves[Fis01] and Perrin’sAlgebraic Geometry: An Introduction[Per08] are all good introductions for graduate students.
Another good place for a graduate student to get started, a source that we used more than once for this book, is Kirwan’s ComplexAlgebraic Curves[Kir92]. Kunz’sIntroduction to Plane Algebraic Curves[Kun05] is another good beginning text; as an added benefit, it was translated into English from the original German by one of the authors of this book (Richard Belshoff).
Holme’sA Royal Road to Algebraic Geometry[Hol12] is a quite good recent beginning graduate text, with the second part a serious introduction to schemes.
本书单摘自：Algebraic Geometry: A Problem Solving Approach (AMS Student Mathematical Library) Student Edition
by Thomas Garrity (Author), Richard Belshoff (Author), Lynette Boos (Author), Ryan Brown (Author), Carl Lienert (Author) etc.
的前言。
高中生可看的代数几何书Algebraic Geometry:A Problem Solving Approach pdf+Solution
http://www.mathsccnu.com/forum.php?mod=viewthread&tid=3936这是本初级的入门书，是11个美国数学协会的代数几何专家合著的学生教科书。本书前面3章就是讲高中解析水平就可以看懂二次、三次曲线。后面的几章也只需要大学的抽象代数知识。