What problems, ideas or questions first got you interested in algebraic geometry?
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?
Solution 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.
Solution 2:
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.
Solution 3:
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.
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.