# Algebraic Curves reference

I am an undergraduate student. I finished a Galois theory course. I started to read Fulton`s book Algebraic Curves.I am doing self-study. I found it difficlt to understand from 2nd chapter onwards. The notes I find are too concise. Are there some lecture videos, or expanded notes.so that I can understand it easily?

-

Algebraic geometry is a hard subject, and Fulton's book is one of the easier ones out there. Possibly Undergraduate Algebraic Geometry by Miles Reid might be more accessible, and you could take a look at that.

But really, Fulton is a good introductory book. It's not a sort of grab-bag intro like Reid, but an actual, honest to goodness, introduction to commutative algebra and the algebraic geometry of curves. However, you are right that it is very dense. Unfortunately, there is no "easy way" to learn math. You simply have to work through it. The best thing you can do is go slowly and post any questions you might have on here. If you did well in a course on Galois theory, I have no doubt you can understand Fulton's book if you put your mind to it.

-

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.

-