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I've just finished teaching a year-long "foundations of algebraic geometry" class. It was my third time teaching it, and my notes are gradually converging. I've enjoyed it for a number of reasons (most of all the students, who were smart, hard-working, and from a variety of fields). I've particularly enjoyed talking with experts (some in nearby fields, many active on mathoverflow) about what one should (or must!) do in a first schemes course. I've been pleasantly surprised to find that those who have actually thought about teaching such a course (and hence who know how little can be covered) tend to agree on what is important, even if they are in very different parts of the subject. I want to raise this question here as well:

What topics/examples/ideas etc. really really should be learned in a year-long first serious course in schemes?

Here are some constraints. Certainly most excellent first courses ignore some or all of these constraints, but I include them to focus the answers. The first course in question should be purely algebraic. (The reason for this constraint: to avoid a debate on which is the royal road to algebraic geometry --- this is intended to be just one way in. But if the community thinks that a first course should be broader, this will be reflected in the voting.) The course should be intended for people in all parts of algebraic geometry. It should attract smart people in nearby areas. It should not get people as quickly as possible into your particular area of research. Preferences: It can (and, I believe, must) be hard. As much as possible, essential things must be proved, with no handwaving (e.g. "with a little more work, one can show that...", or using exercises which are unreasonably hard). Intuition should be given when possible.

Why I'm asking: I will likely edit the notes further, and hope to post them in chunks over the 2010-11 academic year to provoke further debate. Some hastily-written thoughts are here, if you are curious.

As usual for big-list questions: one topic per answer please. There is little point giving obvious answers (e.g. "definition of a scheme"), so I'm particularly interested in things you think others might forget or disagree with, or things often omitted, or things you wish someone had told you when you were younger. Or propose dropping traditional topics, or a nontraditional ordering of traditional topics. Responses addressing prerequisites such as "it shouldn't cover any commutative algebra, as participants should take a serious course in that subject as a prerequisite" are welcome too. As the most interesting responses might challenge (or defend) conventional wisdom, please give some argument or evidence in favor of your opinion.

Update later in 2010: I am posting the notes, after suitable editing, and trying to take into account the advice below, here. I hope to reach (near) the end some time in summer 2011. Update July 2011: I have indeed reached near the end some time in summer 2011.

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    $\begingroup$ Dear Ravi, while I'm not sure if this should be taught in a first schemes course, but it's something that I'd love to see exposited more fully. Jim Borger gave an outline of a program to jump straight into algebraic spaces, skipping schemes entirely. Maybe you could figure out a way to do it? sbseminar.wordpress.com/2009/08/06/… $\endgroup$
    – Harry Gindi
    Commented Jun 17, 2010 at 12:57
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    $\begingroup$ It's also worth linking to the meta discussion that Ravi started to help him craft this question before asking it: tea.mathoverflow.net/discussion/446/… $\endgroup$
    – Andrew Stacey
    Commented Jun 17, 2010 at 14:30
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    $\begingroup$ A parsing question: does "first serious schemes course" mean that there could be a prior, not-so-serious course on schemes? Or do you mean "first, serious schemes course"? $\endgroup$
    – Pete L. Clark
    Commented Jun 17, 2010 at 17:15
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    $\begingroup$ Hi Pete, that was deliberately vague. There's a strong case for a varieties course before a schemes course. And the varieties course could be constructed so that schemes are secretly being set up. (I tried to do this once, when teaching a one-semester course at MIT.) Mumford's Red Book is great at this --- the way he does varieties makes schemes quite natural. But what I really meant was "a first schemes course" and "a serious schemes course" separately: not a second course where you see fancier things, nor a light introduction to schemes, nor a varieties class. $\endgroup$
    – Ravi Vakil
    Commented Jun 17, 2010 at 23:40
  • $\begingroup$ You say the students are "from a variety of fields". Did you mean various fields within mathematics or from other fields than mathematics? If the latter, I wonder which fields outside of mathematics make use of algebraic geometry? $\endgroup$
    – Michael Hardy
    Commented Dec 18, 2021 at 2:05

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As David and Anweshi told before, think it could be very interesting to deal with functor of points, with main example being subfunctors of Grassmannians. I would make some general statements on functor of points (Yoneda lemma, definition of functor of points, vector bundles) and then begin to study as soon as possible classical examples, such as Grassmanians, Severi-Brauer varieties and their tautological vector bundle, varieties of flag of subspaces...

Finally it would lead to a glimpse on group schemes and algebraic groups.

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  • $\begingroup$ This sounds sensible to me. I've managed to get to these things later than I would have liked, largely because I didn't make a bee-line for them (notably Grassmannians) from the start. $\endgroup$
    – Ravi Vakil
    Commented Jan 3, 2011 at 23:03
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Being a differential geometer, it might be nice if you can point out analogies (perhaps even make them rigorous ?) to differential geometry. Like a scheme being flat over another is perhaps akin to a fibre bundle. A scheme itself is like a manifold, etc. This might make the subject slightly less scary for geometric analysts.

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Scheme theory is abstract. I think motivating and concrete examples are important, e.g. elliptic curves and their arithmetic (Weil conjectures for elliptic curves, for example, or other topics from the two books of Silverman on elliptic curves).

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