I was just wondering what the real prerequisites are for reading Qing Liu's 'Algebraic Geometry and Arithmetic Curves', and if it is a good first book on the subject. In his preface he states that the prerequisites are few and any graduate student possesses the background necessary to read it, but this being algebraic geometry I am reticent to believe him. For example, does he assume knowledge of Differential geometry? Algebraic Topology? I expect that he assumes commutative algebra, but at what level? Anyways, for people that have read him, please share your thoughts/comments on this.
No differential geometry or algebraic topology is necessary, though the former might help with motivation for some things, depending upon your mathematical inclinations. You need to know basic graduate abstract algebra (he develops most of the commutative algebra he needs so you don't need to know that in advance) and you need to know some basic point set topology (definition of a topological space, continuous map, Hausdorff, not much more). That's it. For serious.
Would it help to already know some commutative algebra? Yeah, probably, but it's not essential by any means. Liu's is a remarkably self-contained book. I consider it to be an excellent first book on modern algebraic geometry. Whether or not you should first learn some classical algebraic geometry depends on you and your tastes. Liu does do some of that (algebraic sets) in the second chapter. Being familiar with the classical theory might help with motivation, but I don't think it's necessary.