How to learn algebraic geometry - a step-by-step guide

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Algebraic geometry has a reputation for being incredibly difficult and abstract.

And that’s no accident — it is quite a difficult subject to get into!

An excerpt from Ravi Vakil’s “The Rising Sea”

In this article, I want to share with you all the resources I wish I had when I was learning the subject.

There are two main perspectives to the subject:

1. Elementary algebraic geometry. This is the study of curves, varieties, and projective space. This can be learned using high-school-level algebra and a bit of abstract algebra, namely ring theory and group theory.

2. Schemes. This was the revolution that Alexander Grothendieck spurred on in algebraic geometry. It is an incredibly powerful language but it is notoriously difficult to learn.

In this article, I’ll share resources to cover both of these things. In my experience, it’s best to first learn 1) and then go to 2). Otherwise, it’s very easy to get bogged down by 300 pages of definitions about schemes before getting to any actual geometry.

Elementary algebraic geometry

1. “A guide to plane algebraic curves” by Keith Kendig. It’s written in a very elementary style and has lots of really captivating diagrams throughout. If you look at the table of contents, it starts off with lots of examples that only require elementary algebra. And by the end, it actually gets to some pretty deep theorems in algebraic geometry.

   

2. "Ideals, Varieties, and Algorithms” by Cox, Little, O’ Shea. This book does not assume any knowledge of abstract algebra and teaches everything from the ground up. It is a very nice book with plenty of computational examples and exercises.

3. (Shameless plug!) I made a video last year explaining the basics of what algebraic geometry is. I tried to focus as much as possible on the high-level ideas, to avoid getting bogged down by too many details.

Learning about schemes

There are two books that I really enjoyed here.

1. “Geometry of Schemes” by Eisenbud and Harris. I love this book. It’s full of examples, pictures, and exercises, making it really good for self-study.

   

2. “Algebraic Geometry and Arithmetic Curves” by Qing Liu. This book is a graduate level is all about schemes and Spec. It's a rather terse theorem-proof style book, but it is beautifully written and has lots of exercises.

Problem of the Week

As always, here is this week’s challenge problem.

In how many ways can you tile a 3×n rectangle by 2×1 dominoes?

If you have a solution to the above challenge problem, submit it here for a chance to be featured in the next issue of this newsletter.

See here for the solution to last week’s problem. (Shoutout to Matvei Zhukov from Finland whose solution is being featured this week!)

Until next time,

Adithya