# Help understanding Algebraic Geometry

I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with algebraic or projective geometry.

I'm wondering if any of you out there know of any articles, blog posts or whatever offering a light, intuitive and geometric introduction the subject. I really wanna get back to Hartshorne's book cause I am very curious about the categorical description.

I have provided the first few problems I ran into to give you an idea of where I come from. Of course if you can answer any of the questions that would be welcome.

First of all I'm having trouble grasping the very basic notion of a continuous function with respect to the Zariski topology. I don't which they are or know how to conceptualize them. I get how the rational polynomials work but I don't know if they are a subclass of the continuous functions or if they exhaust them. Any help in this regard is welcome.

Further I couldn't really get the projective part. I guess part of my problem comes from the fact that this is a set theoretic quotient of an algebra, which is then interpreted as an algebraic object. At least that's what I read, might be wrong. I seem to get lost during this transition and I don't know how to relate, are there any universal properties involved, whats the big picture?

Edit1: Also, where is the hyperbolic geometry in all this?

Edit2: I want to express my gratitude towards all the people who have takes their time to give me recommendations and sympathy. Thank you!

• Hartshorne's is a great book to have a reference and/or to study from after one already have some (in fact, some serious) background in algebraic geometry, but it is the worst book for a beginner. You're not alone in this, many fell in that hole (I, among others...), and it was pretty frustrating until I found out there are way better things to begin with. I recommend Reid's "Undergraduate Alg. Geom." , Hulek's "elementary A. G.", Harris' "A.G.: A First Course" , Bump's "A.G.", etc. Check these in your mathematics dept's library. Jan 2, 2013 at 22:01
• If you want light, intuitive, and geometric, it's definitely worth checking out the notes here: mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf. Certainly Chapter 0. (I don't recall how to make nice links in comments, sorry.) Jan 2, 2013 at 23:45
• There are too many continuous functions with respect to the Zariski topology. Nonetheless, the Zariski topology is the coarsest topology that makes rational functions continuous. (We put the Zariski topology on the affine line, of course!) Jan 3, 2013 at 10:46
• I would suggest taking a look at the chapter in William Fulton's Algebraic Curves (available free online) that introduces projective space (Chapter 4 maybe). I didn't understand the point of the construction until I read this. Jan 4, 2013 at 16:20

Recently, the best freely available textbook on category-laden algebraic geometry seems to be:

The following reference is a great companion to the hard core of Vakil and/or Hartshorne:

For deep classical projective algebraic geometry, I cannot but eagerly recommend:

For a mixture of both, with a first half introduction to projective algebraic geometry and a second half heavily focused categorical introduction to schemes, this new book is a gem, and may be exactly what you are looking for, serving as a perfect introduction before/along with Hartshorne first chapters:

You can get part of the scheme theory of that book for free at Holme's website. Definitely, Holme's book will be more than enough (maybe along with Gathmann's notes, see links below) to fill in geometric motivations for Hartshorne; jointly with Vakil's course complementing the categorical side, you will have enough and almost self-conteined material to digest for a long time.

Besides the recommendations given already, I would suggest you check out the other useful posts I referred to in this other answer. The lecture notes by Kerr provide a lot of geometric motvation and intuitive pictures on projective algebraic curves, and Gathmann's thorough course gives a highly insightful and motivated broad introduction to the more abstract approach, being an excellent detailed "overview" before approaching Hartshorne (as the author himself points out in his bibliography).

To clarify concepts on projective geometry, projective varieties and to supplement Hartshorne's reading, either from a complex geometry or purely algebraic point of view, the following long list of freely available online courses may provide you with the extra bits you need on specific topics (warning! most of them are more elementary than Hartshorne but some of them go beyond it or supplement it on other topics, they are included for completeness of good references to have if you decide to go beyond Hartshorne):

• I managed to get my hands on A Royal Road. Havent read much but from the content it seems like its just right. I'm gonna go with the authors recommendation and read 5 pages every day then stop and look at the view. Thanks! Jan 4, 2013 at 10:13
• @user25470: thanks! check out the new list of free online courses I just added, you may get good insights and supplementary explanations/examples/exercises there. Anyway, Holme is a very good new book for an rigorous introduction before Hartshorne. Jan 4, 2013 at 11:08

See the Algebraic Geometry site of Donu Arapura's from Purdue University, where you'll find links to:

• You might also want to check out J.S. Milne's site on Algebraic Geometry, where you'll find a list of content that's covered, and a downloadable pdf file.

• There's an "online" course website, for a class on Algebraic Geometry at Stanford University, Foundations of Algebraic Geometry, where you can access support material, including Notes compiled by R. Vakil.

• I read through half the prequel, it had alot of nice examples, thanks Jan 4, 2013 at 9:53
• Your welcome, user25470. I refrained from throwing a huge list of references because I wanted to point you to resources that seemed best to meet your needs, as expressed in your post. Many of the suggestions in the huge list in the answer above will likely not help you NOW to prepare for or understand Hartshorne (as many go beyond Hartshorne); perhaps those references will be useful to you down the road, if you pursue algebraic geometry. Jan 4, 2013 at 14:24
• Yeah I know. The reason I accepted his post was that I again got lost when I came to the projective part. While most of the references seem to be above me the royal road seems like a fairly grounded and intuitive text. Jan 5, 2013 at 12:21
• @amWhy: Time for a nice answer nadge! :-) +1 Apr 30, 2013 at 1:04

When starting out I found Klaus Hulek's Elementary Algebraic Geometry (Student Mathematical Library Vol. 20) to be great. The beginning might be a little rough, so glance through it with the idea of coming back to it as needed, but once it gets into the main topics he does tons of very concrete examples (with equations and everything!) to give a good feel for what is going on.

I found projective geometry confusing when I began learning algebraic geometry. Hartshorne has notes on projective geometry which are available online and which I found quite useful. Search Foundations of Projective Geometry by Robin Hartshorne online, or contact me via email (you will find my email address on my profile) and I will send you the notes.

However, I am not sure if Hartshorne's notes will be too elementary for you. Also, here is a link to a webpage for an REU I did last summer, and our supervisor wrote some nice notes which you may find useful: http://math.columbia.edu/~dejong/reu/doku.php

You may also want to take a look at Joe Harris's Algebraic Geometry: A First Course. He has many examples, especially with projective stuff and explains the Zariski topology. In my opinion, this is not an easy book to read, even though the title says it is a first course.

You can download the whole book titled "Algebraic geometry over the Complex numbers" by Donu Arapura at the following link.