What problems, ideas or questions first got you interested in algebraic geometry?

Question

Obviously, a lot of people are very interested in algebraic geometry. I suppose this means it is a fascinating area. However the few times I have tried to read introductory books and/or articles in the area, I haven't been able to "get it" or even to see what all the fuss is about. It seems pretty technical.

A few questions for those who are involved in algebraic geometry:

• What was your first exposure to algebraic geometry, and did you immediately like it? If not, what made you go back and study it later?
• What problems, ideas or questions first got you interested in the field?
• Can you recommend a strategy for breaking into the field such that the material will seem as well-motivated as humanly possible?

Answer 1

I went through a bunch of geometry-type courses in college (physics, differential geometry, parts of algebraic topology, etc.) and really just had absolutely no idea what anyone was talking about. People would give very non-rigorous arguments for things and I'd end up having no idea quite what they were actually saying. Sure, it sounded reasonable when they described things, but when I tried to do the same things I'd end up with four different answers and no idea which was correct.

So for several years I gave up on geometry entirely, secretly harboring severe doubts as to how much of it was even true.

Years later, I was in grad school with the idea of doing something purely algebraic and totally unconnected to geometry. I had some free time and decided to stop by the library. Occasionally I'd do this, picking out a math book at random to see if I could get some idea of what it was about. On this occasion, I picked up Daniel Perrin's Algebraic Geometry. (For those who aren't familiar, the book is unlike a lot of others in that it starts out with an actual proof of the Nullstellensatz, instead of assuming the reader is already familiar with commutative algebra). For the first time in my life, I was seeing geometric arguments where I could actually check each step and satisfy myself that they made sense; there was no ambiguity about what was being done, I wasn't asked to "imagine deforming such-and-such a little bit, which will obviously have a non-negligible effect on this and a negligible effect on that..."

Having this tiny foothold let me do a number of things. For one thing, I could see the geometric ideas motivating a lot of things I've previously thought of as miraculous formal tricks. (I'm pretty sure that, up to that point, I hadn't even understood that the Fundamental Theorem of Calculus was something other than clever pencil-pushing!) For another, I could go back and now see what people meant when they described geometric arguments informally.

I ended up transferring to a different grad program when I was nearly A.B.D. and just almost starting over. I'll be older than I'd like by the time I'm applying for postdocs, but I'd do it again; it was worth it to understand the geometry. I just wish I could have learned all this a little bit earlier and avoided wasting so much time.

Regarding "getting started," I recommend the following:

• Daniel Perrin, "Algebraic Geometry - A First Course"
• Karen Smith et al., "An Invitation to Algebraic Geometry"
• Cox, Little, and O'Shea, "Ideals, Varieties, and Algorithms" -or- Brendan Hasset, "Introduction to Algebraic Geometry"
• Paolo Aluffi, "Algebra: Chapter 0"
• Rick Miranda, "Algebraic Curves and Riemann Surfaces"
• David Mumford, "The Red Book of Varieties and Schemes"
• David Eisenbud and Joe Harris, "The Geometry of Schemes"

There are a couple of things you'd need to learn at this point that I've never seen written down in a way I consider satisfactory. These include how to prove things about schemes (first, reduce to a local commutative algebra problem, then quote some deep theorem from commutative algebra you've probably never heard of before); how to work with vector bundles; the general ideas of homological algebra; etc. These are all things I tried to learn out of books forever and only ever learned once I had someone who could explain them to me.

Answer 2

I would say that understanding the Nullstellensatz is the first step of anything in algebraic geometry. It could be also possible to start doing pure commutative algebra for a while and then do the Nullstellensatz, but the other way around is possible too. A course in commutative algebra would be the best, but I personally liked Atiyah-MacDonald for commutative algebra basics and Dummit & Foote's Abstract Algebra for the basics of algebra (elementary ring theory and module theory). There I got interested in generalizations of number-theoretic ideas to purely algebraic statements, such as the Chinese Remainder Theorem or the classification of finitely generated modules over a PID/Dedekind Domain, which has several applications and is interesting in its own right.

Hartshorne's book is definitely one you want to get into at some point, but not at first, mostly because it's a bit intense and relies way too much on you doing the exercises, so if you don't have enough commutative algebra to do them, you'll feel the book is really harsh. I suggest you do go through this book one day as it is the most standard reference, but it will take you a while to read (I'm saying "a while" to not say "months"). But the heart of modern algebraic geometry is hard to learn and there is no way around it.

A way around the hard things though is to look at the statements of algebraic geometry and prove them in particular cases, for instance studying classical varieties over $\mathbb C$ ; this is definitely something you want to do before going into abstract algebraic geometry and "break into the field". The best book for this in my opinion would be Joe Harris' "Algebraic Geometry". This book is known for doing lots of examples and dealing with problems "by hand", so to speak, so that you can see in the proof what's going on, instead of using super general arguments which are very powerful but very abstract too.

For instance, Harris proves that a morphism between two (classical) projective varieties is a closed map by using results on resultants and linear projections from a point, where as in Hartshorne, they show that a projective morphism between Noetherian schemes is proper (hence a closed map in the classical case) using what they call the "valuative criterion for properness", which is still kind of obscure to me.

Hope that helps,

Answer 3

When my first contact occurred I didn't even know that it was algebraic geometry because the course was called "analytic geometry"!
It was in high-school when I was about fifteen years old.
I liked algebra and geometry and it was fascinating to discover that geometric figures that I had learned about in the Euclid style could also be described by algebraic equations which could be manipulated in a mechanical way.
Also I had the impression that calculations with these equations gave more more rigorous arguments than the Euclidean-style proofs I had been exposed to .
As an aside, I think that young mathematicians would be amazed at the wealth of pretty and rather deep results in courses that were then called analytic geometry in European secondary schools (and probably elsewhere: I don't know).
Of course foundational questions were avoided and I guess it was tacitly assumed that the base field was $\mathbb R$ ( although words like "field" were never pronounced and I had no idea that the word "set" or "group" had a mathematical meaning).
There was no clear-cut distinction between affine and projective plane: suddenly points in the plane that had had two coordinates were described by three homogeneous coordinates and deliciously mysterious tangential coordinates were attributed to points of what we didn't then call the dual projective plane.
But all this was undoubtedly genuine algebraic geometry, terminology notwithstanding.

A reason for this answer
My main motivation for this post is to remind beginners that algebraic geometry is not a conspiration of 1950's geometers scheming (!) to flood innocent mathematicians under spectral sequences and derived categories, but was invented by a brilliant seventeenth polymath, René Descartes, in an appendix to his celebrated philosophical treatise Le Discours de la méthode.
Algebraic geometry is a branch of geometry where pleasantly concrete calculations can and should be done.
Sophisticated methods must eventually be learned in order to solve difficult problems but we should always keep in mind that this sophistication is a means and not an end.

Answer 4

1. I took a 1 year course in abstract algebra (groups, rings, field), then a 1 semester course in commutative algebra (CA), where we covered all of Atiyah-MacDonald (AM). Lot of it was with a view towards algebraic geometry (AG), so some topological things about $\mathrm{Spec} \ A$ were mentioned, and the definition of the structure sheaf (as it is done in the exercises in AM). There I learned about the Nullstellensatz and the Hilbert basis theorem, which are foundational to classical AG.

2. A 1 semester course in AG, covering lots of classical AG (as e.g. in the books of Hartshorne, chap. 1, and Harris), and a bit of scheme theory (Hartshorne, chap. 2). This is when I took the connection between CA and AG more seriously, in particular the viewpoint that "CA is for AG what calculus is for differential geometry" (local analysis).

I can't say that at this point I loved AG. I always felt that in classical AG there is strong dichotomy between the affine and the projective world, I still prefer the affine world (=the category of reduced finite type $\mathbf{C}$-algebras in classical AG) as its general framework (e.g. the importance of the rings of functions) seems to be more satsifying, albeit I find projective geometry rather interesting. I would recommend (as I did) to make the connection to schemes as fast as possible, contemplating Mumfords picture (see here, it requires some clarification, and many connections are not obvious at first sight) of $\mathrm{Spec} \ \mathbf{Z}[x]$, which I found amazing as it unifies geometry, algebra, and arithmetic. When I saw it I was totally amazed, and it made me very eager to learn about schemes. It was also extraordinarily interesting and inspiring to read the introduction to Grothendieck's EGA (Grundlehren edition), it contains so many deep ideas and ways of thinking. It tells you about AG "at large", which Hartshorne also attempts, but does not even come close to it. Apart from that I always wanted to understand a proof of de Rham's theorem. As I was too lazy to read the long differential geometric proof in Lee's book, I learned enough sheaf theory to understand the very short sheaf-theoretic proof of it.

Answer 5

After my master thesis in logic I wanted to find a subject for a PhD Thesis. I asked professors and obtained a sequence of books in different subjects. One was "Undergraduate algebraic geometry" of Miles Reid. After this, I decided to choose this subject for a PhD Thesis.

I definitely recommend this book for readings. It is very easy, you do not need anything to discover the fields and are directly into it. For me the best to start.

Answer 6

For me it was plane curves. At school, the more of them I met, the more I wanted to meet. You change the polynomial a bit and the curve changes in a surprising, often unpredictable way.

I love old books on geometry, like Coolidge Plane Algebraic Curves, Enriques-Chisini Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. And there are newer ones, of course, like Harris Algebraic Geometry First Course, Clemens Scrapbook of Complex Curve Theory.

From geometry comes the mystery and the richness, like interesting behavior, strange scenarios that can occur.

And then we need to efficiently manage the information we are gathering about our geometric objects and make sure our thinking is tidy. At the same time, we'd like to have freedom to express and imagine interconnections between our geometric objects. This is what Grothendieck's machinery is there to help with.

Hartshorne didn't work out for me. I used Mumford's Red Book in the beginning.

By the way, there is a very good new book out by Bosch, called Commutative Algebra and Algebraic Geometry. Bosch can save quite a bit of time, imo.