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
Reference request for algebraic geometry
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You are right, I think, that Chapter 2 of Fulton's can give you a hard time when you read it for the first time, because of the commutative algebra involved. I also learn AG from Fulton's book at the moment and as I already finished Chapter 2 (in fact almost all of the book, yet I have to do all the exercises), so I might give you a rough guideline what Chapter 2 is about, and what in my opinion is important and what is very important.(WARNING: Purely subjective comments by a non-expert in Algebraic Geometry)
Sections 2.1-2.4 are mandatory, in particular section 2.2 on polynomial maps. Do all the exercises of section 2.2, in particular the ones containing concrete examples of varieties (i.e. 2.8, 2.12 and 2.13). The result of exercise 2.7 is very important to keep in mind, since it gives a convenient criterion for an affine algebraic set to be a variety.
Section 2.4 is also very important, because it contains the definition of the local ring at a point of a variety. This concept is absolutely fundamental in Chapter 3, where intersections of affine plane curves are studied via local rings (roughly spoken, the way two plane curves intersect at a point is completely encoded in the local ring at that point, modulo the ideal generated by those two curves).
Section 2.3 is purely technical. It is of importance in Chapter 3 as well, because the results obtained in that section allow you to reduce most arguments involving an arbitrary point $P \in \mathbb{A}^{2}(k)$ and two distinct lines $L$, $L'$ passing through $P$ to the case, where $P=(0,0)$ and where $L$, $L'$ are the two coordinate axes (see Exercise 2.15d).
The rest of the sections are pure algebra with algebraic geometry in hindsight. So I'll give you just a few comments on each section
Section 2.5: The point why DVR's (discrete valuation rings) are so important is the fact that the local ring at a simple point on a plane curve is a DVR (Theorem 1, section 3.2), and as the local study of plane curves involves a detailed study of the local rings of that curve, it is nice to know that most such local rings are rather simple (i.e. have only one maximal ideal, which is even principal). You won't need much of the stuff discussed in the exercises until Chapter 7, but for the exercises of Chapter 3 you definitely need Exercise 2.29. So do at least this one.
Section 2.6: This can be postponed until you finished Chapter 3, since you will need it only from Chapter 4 onwards (but you really, really need it from that point on!).
Section 2.7: Short and painless. It's so short you should read it.
Section 2.8: Here the exercises are far more important than the text. Do Exercises 2.42-2.46, they are all very important in Chapter 3 and beyond.
Section 2.9: This basically contains only one important result, so keep the statement in mind, but I don't think you need to understand all of the proof in order to carry on, so you might wish to skip a detailed study of the proof for now and do that later.
Section 2.10: This is a very important section, because some proofs of important theorems (such as Bezout's Theorem) are proved by counting dimensions of certain vector spaces, and by comparing them via short exact sequences. So although the exercises may be tiresome and formal doing all of them is very important (Fulton sometimes uses results from those exercises without explicitly referring to them all over the book).
Section 2.11: This is material used in Chapter 8, so you might skip it on your first read.
Lastly a word on the exercises scattered throughout the text: Fulton uses a whole lot of those exercises in the main text. Most of the time he is giving an explicit reference. But the result of some exercises might prove extremely helpful in solving other exercises, and Fulton may or may not tell you which of the results is important in solving this exercise. So you should really at least try to do all of them. Otherwise you won't get the best of the book and you will have to settle for much less.
Hope I could help you. And any criticism (in particular of experts of algebraic geometry) is of course very much welcome.
Here are a few options:
Real Algebraic Geometry by Bochnak, Coste and Roy.
This seems to be the standard reference for Real Algebraic Geometry. Most of the chapters(at least the first 5) should be accessible with a bit of work. Later chapters will require a bit more background.
Introduction to the Real Spectrum by P.L. Clark
This gives an overview of some of the ideas behind real algebraic geometry. It starts by defining what ordering of rings are and how they connect to geometry. Moves on something called the real spectrum of a ring together to results related to it.
Introduction to Real Algebra by T.Y. Lam
This is one my favorites intro papers. It is clearly written and presents the material well. It is a bit dated but I like how it treats the Real Spectrum. Might be a bit too advanced, specially if you have never seen scheme theoretic approach to algebraic geometry.
Real Algebraic Sets by M. Coste
Not sure if this is an intro to the subject but it gives a quick overview of semi-algebraic and real algebraic sets and discusses some topological ideas related to it. For example, how in low-dimension we can characterize real algebraic sets.
algorithmic approach:
Introduction to Semi-algebraic geometry by M. Coste.
Semi-algebraic geometry is often used as a synonym for real algebraic geometry. This gives you a quick intro together with some of its computational tools.
Algorithms in Real Algebraic Geometry by Basu, Pollack and Roy
Similar in spirit to the above, but a lot more comprehensive. Contains a lot of the background material. See also a more recent online version for updates and fixes.
Remark:
You might not be find a complete linear path to learning geometry. I am not sure if this is 100% sound advice but just get stuck in. If you then find some material which you haven't encountered you can take a small detour.
Best Answer
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.