# Varieties as an introduction to algebraic geometry / How do professional algebraic geometers think about varieties

This really is two questions, but they are kind of related so I would like to ask them at the same time.

Question 1:

In a question asked by Amitesh Datta, BCnrd commented that it is important to learn about varieties in a classical sense before learning about modern algebraic geometry because it is where much of the intuition in the subject comes from.

I was hoping to get some opinions on how much one should learn about varieties (in the sense of chapter 1 of Mumford's red book) before moving onto more modern formulations of algebraic geometry.

Is one meant to gain a rudimentary understanding of varieties and then start learning about schemes, OR is one meant to have a really good understanding of abstract varieties before learning about schemes.

Question 2:

Do professional algebraic geometers think about varieties from a scheme theoretic perspective or from a classical perspective.

This is a seriously soft question, so I will make it community wiki. I am however half expecting it to be closed.

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Just a comment, but Ravi Vakil is teaching a full-year algebraic geometry course where he is doing a "schemes-first" approach. The great thing about the notes is that they are written so the reader does not have to check Matsumura every time an important fact from commutative algebra comes up. He's putting the notes online bi-weekly, so you may want to check those out. Regarding the question: There are still modern ways to cover classical algebraic geometry. It turns out that sheaf theory is a much better way to describe even classical varieties than the more classical ad-hoc methods. –  Harry Gindi Oct 21 '10 at 3:22
That is, sheaves are in some sense the universal way to organize local data attached to a topological space. For example, this is true when we look at holomorphic manifolds, where the important sheaf is the sheaf of holomorphic functions back to $\mathbf{C}$. The classical theory of varieties is an abstraction of that idea to the algebraic setting. –  Harry Gindi Oct 21 '10 at 3:28
As I commented here, there are at least three ways that schemes generalize varieties, and it's needless and unfortunate that they are all taught at once. mathoverflow.net/questions/28496/… –  Allen Knutson Oct 21 '10 at 13:21
The question is valuable for bringing out the range of current views on the subject. Having first encountered algebraic geometry in a year-long course based on Weil's Foundations (taught by Tamagawa), I feel disoriented for life and can therefore sympathize with the dilemma faced by others who want to learn the subject in the "right" way but while making some contact with actual geometry. Weil's book, as I recall, has no pictures whatsoever. –  Jim Humphreys Oct 21 '10 at 22:43
Dear Jim, Your recollection of Weil's book is correct. It also spends a lot of time explaining (without using functorial language) that $k[x_1,\ldots,x_n]$ is the free object on $n$ generators in the category of $k$-algebras, and no time explaining the intuition underlying intersection theory. I think it's fair to regard Weil's book as being written for an audience who already understands geometry, with the goal of introducing a rigorous algebraic foundation for the subject they already know (which in particular would work in char. p). In particular, I don't think it was ever intended ... –  Emerton Oct 30 '10 at 21:25

When you are truly fluent in scheme theory, you don't know whether you are "thinking schemes" or "thinking varieties", the intuitions are merged together.

As to learning, for most people starting with schemes is a bad idea, because they don't get to build the necessary intuition, and unmotivated formalism can be quite repulsive; but there are (very few) students with unusually abstract inclinations for whom starting with schemes is just fine.

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I agree that your goals are relevant to this question. I.e. do you want to "learn", "understand", or "use" algebraic geometry?, or perhaps write a thesis? These are all different. If one wants to understand the subject, I like the historical approach, beginning with Riemann surfaces over the complex numbers, say from a book like that of Rick Miranda (augmented by reading Riemann). I.e. I think it is useful when learning an abstract subject to know what elementary things it generalizes, rather than just memorizing the general version.

Of course everyone is different, as my friend George Kempf apparently just sat down and read EGA, but that didn't work for me.

For varieties it helps to supplement Mumford's red book by Shafarevich's Basic Algebraic Geometry. Joe Harris's book Algebraic Geometry derives from his experience teaching algebraic geometry first by concrete examples at Harvard and Brown, but very little theory, which he said seemed to work well.

It is also my view that if you want to use the subject in geometry, including calculate with it, that sheaf cohomology is more important than schemes. Thus George Kempf's book (or Serre's FAC) which treats cohomology of varieties, may be more useful than studying schemes. if you want to do number theory on the other hand, I am assured that schemes are fundamental.

I guess the historical order would be roughly: Riemann surfaces, algebraic curves, algebraic surfaces, general projective varieties, sheaf cohomology of varieties, schemes and their cohomology.....

If you want to learn Riemann surfaces and sheaf cohomology at the same time, Gunning's Princeton lectures on Riemann surfaces are excellent. But there you will only learn the analysis and not the geometry.

I am surely hopelessly naive, but to me schemes are just varieties where you also remember the equations, and stacks are just moduli spaces where you remember the automorphism groups.

A remark: one way to think of schemes is as a "limit of varieties". This is not quite general, but if one has some equations (f1,..,fr) in n variables, they give a map from n space to r space, whose fiber over most points is usually a smooth variety, but whose fiber over the point 0 is more special. That fiber, with its scheme structure, essentially determines the (possible) nearby varieties that may be thought of as converging to the special fiber over 0. Thus knowing the scheme structure can help tell you how the object would change if its structure is slightly varied. E.g. the scheme defined by x^2=0 tells you it is a limit of two points, while that defined by x=0 is a limit of one point.

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I like this answer a lot. –  Emerton Oct 30 '10 at 21:27

I stand by my answer to the question that you linked. In particular, I think that the distinction between "classical" and "modern" algebraic geometry is a little artificial, and I don't think that anyone is meant to do any particular thing; what you need to know depends on what theorems you want to read/use/prove.

But whatever direction you ultimately intend to pursue, it makes good sense to learn varieties first. As well as Chapter I of Mumford, there is Chapter I of Hartshorne, and its many exercises. The first few sections in particular are crucial. There is also Griffiths and Harris, which has no mention of schemes, as far as I can recall, but an awful lot of algebraic geometry of varieties.

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Dear Harry, Yes, one can do this, but there are already complications; if one wants sheaves of functions, but also wants to, e.g., study the $\mathbb C$-points of a variety defined over $\mathbb Q$, one has to worry a bit about what field the functions take values in. Alternatively, one just passes to the scheme picture, but then one has to discuss abstract sheaves, rather than sheaves of functions, which is more baggage. My view is that schemes are an extended metaphor --- and to understand the geometric intuitions for which they are a metaphor, it helps to understand ... –  Emerton Oct 21 '10 at 7:08
... classical varieties first. My feeling is that certain geometric ideas (Bezout's theorem, blowing up, families of varieties, etc.), which are at the heart of algebraic geometry, can be profitably introduced first in the simple context of varieties. Certainly one can use sheaf- and scheme-theoretic ideas to go further in the study of them, but at the beginning, having a lot of technical baggage can (in my view) obscure the simple geometric ideas. Once you understand the ideas, the technical baggage itself becomes much less of a burden, since one then knows its intent and purpose. –  Emerton Oct 21 '10 at 7:13
Harry, in my undergrad differential geometry course at UM I only discussed the idea of a sheaf of R-valued functions (avoiding the word "sheaf"!) as a substitute for atlases, and some related notions for different ways to think about vector bundles. Bad idea to introduce general concepts of sheaves or non-closed points in undergraduate courses. One merit to discussing sheaf of k-valued functions is normalization of non-affine varieties, and requires no non-closed pts or general sheaf stuff, but I tried that experiment once with an undergrad a.g. course and it was very hard for the students. –  BCnrd Oct 21 '10 at 7:17
... I would prefer to do some geometry first. (Sheaves really demonstrate their merit once you start taking their cohomology. But sheaf cohomology is a fairly sophisticated tool, and I would rather introduce and explore some geometric problems, before describing how to solve them using cohomology. For example, arguments with $\mathcal O(n)$ and its cohomology on projective space are much more easily motivated, and understood, once you have some experience with the classical idea of taking intersections with hyperplane or hypersurfaces, of arguing with graded pieces of homogeneous ... –  Emerton Oct 21 '10 at 8:08
That is a good question. I guess I need to learn more geometry! –  Harry Gindi Oct 21 '10 at 14:02

I believe the right answer to this question depends very much on how quickly one needs to get up to speed on schemes (and at this point in mathematical history, stacks).

For me personally, I found learning about complex varieties (from Mumford's book "Algebraic Geometry I: Complex Projective Varieties" and Griffiths-Harris) much more entertaining than learning the basics of scheme theory. This is largely a matter of personal taste, but for some it's already a good enough reason to start with varieties: if it's just more fun for you to work with particular examples, varieties are the place to start.

On the other hand, I didn't have to write a thesis in algebraic geometry and it wasn't until later on in my career that I needed anything about schemes--at which point I had the luxury of plenty of practical experience with varieties in characteristic zero to rely on. I can't overstate how much I relied on my experience with algebraic varieties when I began learning about schemes: I could appreciate the added flexibility schemes provide and I already had a zoo of examples under my belt. Without earlier experience with varieties I would have found myself yawning through the mountains of routine but necessary scheme-preliminaries.

But schemes are really an indispensable part of modern mathematical language if you are in one of the many fields that rely on algebraic geometry, and it may therefore be essential to learn about them immediately in parallel with varieties. I don't envy the modern graduate student who has to decide on a tipping point between classical algebraic varieties and schemes/stacks!

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As might be obvious from my answer, I'm not a professional algebraic geometer, so I didn't answer question 2. But I have to admit that despite my preference for varieties, even I occasionally think of them as a special class of schemes. Still, it's important to keep in mind that additional powerful tools are available when one is working over the complex numbers, and thinking of varieties as schemes is not always the best route... –  Sheikraisinrollbank Oct 21 '10 at 11:43

I don't seem to have the option to edit my answer, but I wanted to correct myself: it is the scheme structure of the limit that determines the possible nearby varieties in the family, not vice versa. The question of whether the nearby varieties determine the limit is that of Hausdorfness of the parameter space. One has to restrict the nature of the possible limits to get uniqueness of the limit. This arises in deciding just how complicated the limit should be when trying to construct a nice compactification of a given family of nice varieties. It is a wonderful fact that for smooth curves one can compactify them without introducing non varieties. All one needs is some simple singular curves. This was first noticed by Alan Mayer and David Mumford in their talks at the Woods Hole conference 1964, whose notes appear on James Milne's webpage at Michigan, (and mine at UGA). I am roy smith, and have been using mathwonk as alias for so many years online I have forgotten it is not my name. Is it sufficient to register and add it to my profile, or do I need to sign posts here as roy smith? That would apparently unlink me from all my mathwonk activity so far.

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"and mine at UGA" is a rather unhelpful reference at this point in time. –  Peter McNamara Oct 31 '10 at 16:04
Roy: For what it's worth, both mathwonk's link to the same profile, which links to your website. You've probably just hit some obscure bug. –  arsmath Nov 1 '10 at 14:23
Roy, I deleted my earlier comment. I am not an expert on how things work here, but perhaps the act registering resolved the earlier problems. –  Donu Arapura Nov 1 '10 at 14:31
You were right Donu. A moderator helped me merge the two mathwonks. Thanks everyone. –  roy smith Nov 1 '10 at 21:39