Reference request for algebraic geometry

Question

I'm an undergraduate student; I must read Mumford's Red Book (the first chapter) in order to write the dissertation. However, I find very difficult to understand all the proofs and the propositions of that book; consider that I'm completely new to algebraic geometry, although I know Galois theory and some algebraic topology. I would like a book (or something else) that is more introductory, in order to give a sense to what I read in Mumford's book. I mean, I would like a book that explains the motivation that led to the definitions and the structures of algebraic geometry. Thank you in advance

Answer 1

I can only recommend these lecture notes by Andreas Gathmann. He starts with the classical theory of zero sets and motivates basically all definitions and results by examples or heuristics. After recalling/introducing the classical theory he continues with sheaves and schemes in the same way up to chern classes at the end, which makes these notes one of the best sources to start studying algebraic geometry with (at least imo). What I also like is that they are not too big. For example, the rising sea by Ravi Vakil is also very good, but contains a lot more information and hence needs more time to study through it.

Moreover, since quite many people use these lecture notes point you will probably also be able to find hints or solutions to the exercises if needed. Otherwise you can of course also just make another post here.

Answer 2

This is (arguably) the most elementary book for learning algebraic geometry I have come across:

Algebraic Geometry: A Problem Solving Approach by Garrity et al.

It is the collection of the material used in the Park City Mathematics Institute’s Undergraduate Faculty Program on Algebraic and Analytic Geometry, 2008.

You can find more details here:

https://bookstore.ams.org/stml-66