Material in a first course in algebraic geometry?

First I would like to say that my question is not about what books to use in algebraic geometry; for this there are many threads that discuss this on Math.SE and on MO. My question is about what material should be included in a first course in algebraic geometry. Let me explain.

My university does not offer courses in algebraic geometry - thus I have found the need to try and "create one" by doing a reading course with a lecturer. Now one may say that we should use this or that book, but from experience it is not the book that matters but ultimately the material that one learns. When learning algebraic number theory, I found myself looking at things from Marcus' Number Fields to KCd's notes, to Neukirch, etc.

My question is: What should be included in a first serious course in algebraic geometry? The level of such a course should be for someone who has studied commutative algebra, algebraic number theory and algebraic topology. Preferably, each answer should include a list of "canonical topics" to be studied.

Thanks.

Solution 1:

Personally, I think your goal should be to try to get to Ravi Vakil's book Foundations of Algebraic Geometry as quickly as possible. But since he starts with schemes, it is a good idea to get some familiarity with the classical theory of algebraic varieties.

First, you should learn the basic dictionaries ($k$ an algebraically closed field):

\begin{align} \left\{ \text{regular functions on affine space $\mathbb{A}^n$} \right\} & \longleftrightarrow k[x_1,\dots,x_n] \end{align}

\begin{align} \left\{ \text{points of $\mathbb{A}^n$} \right\} & \longleftrightarrow \left\{ \text{maximal ideals in $k[x_1,\dots,x_n]$} \right\} \end{align}

\begin{align} \left\{ \text{subvarieties of $\mathbb{A}^n$} \right\} & \longleftrightarrow \left\{ \text{prime ideals in $k[x_1,\dots,x_n]$} \right\} \end{align}

\begin{align} \left\{ \text{algebraic subsets of $\mathbb{A}^n$} \right\} & \longleftrightarrow \left\{ \text{radical ideals in $k[x_1,\dots,x_n]$} \right\} \end{align}

You should also learn the similar dictionary for any affine variety $X$ corresponding to a radical ideal $I$ with coordinate ring $k[X] = k[x_1,\dots,x_n] / I$. As part of this, you'll want to learn about the Zariski topology and you'll need to understand the various forms of Hilbert's Nullstellensatz.

You'll also want to learn what projective space and projective varieties are and learn the analogous dictionaries in that setting. Finally, you'll want to know what a quasi-projective variety is.

You'll need to learn what morphisms (also called regular maps) are in these settings. If you understand the category of quasi-projective varieties (both objects and morphisms), you're off to a good start.

You should also get some familiarity with the function field of an algebraic variety and understand the distinction between rational maps and regular maps, as well as between birational equivalence and isomorphism.

Then it may help to see some basic geometric constructions in a classical setting (Zariski tangent space, singularities, divisors), though you can learn this later if you are willing to accept on faith that quasi-projective varieties (and their generalization to schemes!) are worthwile geometric objects to study even though you don't yet have many tools in your geometric toolbox. Also, the first section of Shafarevich's book has a nice sampling of the types of problems that algebraic geometers are interested in, so it's definitely worth reading, though not necessarily for mastery at this point.

Solution 2:

I spent a lot of time researching this question before teaching algebraic geometry this fall. The goal here is a one term (3.5 month) course from the ringed space perspective that would prepare people for a second course on schemes using Hartshorne or Vakil. I assumed prior exposure to commutative algebra, but not mastery, and no analytic background. I figure I'll share my research here.

I started by brainstorming what I thought was a minimal list of topics:

1 The correspondence between ideals and varieties

2 Localization

3 Projective geometry

4 Classical examples (Grassmannians, flag manifolds, discriminants, etc)

5 Finite maps and conservation of number

6 Blow ups

7 Dimension theory

8 DVRs and local geometry of curves

9 Differentials and derivations, regularity. I later discovered that many courses split this into 9a: Zariski tangent and cotangent space and 9b: Global properties of 1-forms.

10 Normality and normalization

11 Global geometry of curves

I should mention that this list is inspired by an amazing course I took from Brian Conrad which; as I recall, covered all of these except 4 and 5, plus sheaf theory. Conrad has notes from a similar course online here but I found it difficult to verify (or refute) my memory from his notes.

I then went through a number of courses I could find by instructors I respected to see what they covered. Here a minus sign means that a topic seems to have been touched on briefly but not in a detailed way.

Karen Smith 1 2- 3 4 7 9a 6 8 10- 11- 9b Also does Weil and Cartier divisors.

Dragos Opera 1 2- 3 7 9 10 6 5- 11-

Ravi Vakil 1 2- 3 7 9a 8 10 11 9b Also does non-discrete valuations and completions

Igor Dolgachev 1 2- 3 5- 10- 7 4 9 11

Shavarevich's textbook 1 3 7 4 2 9a 6 10 9b 11

Milne's textbook 1 2 7 9 3 4- 10 5 6

What I got from this: My list is too large to cover in a term; you need to choose a subset of this. Everyone seems to agree that 12379 is the core, and almost everyone goes in that order. Almost no one does what I would consider a good coverage of 5. (Sample challenge: Suppose that I hand you the computation that the Fermat cubic has $27$ lines and that the corresponding correspondence variety in $G(2,4) \times \mathbb{P}(\mathrm{cubics})$ is smooth over the Fermat cubic. Do the students know enough to deduce that the generic cubic surface has $27$ distinct lines? For a classical enumerative algebraic geometer, that is the point!) 11 is frequently chosen as the climax of the term.

My own course in progress has day by day notes here. If all goes according to plan, I will cover

1 2- 3 5 4- 7 9 8 10-- 11

Solution 3:

Ravi Vakil once taught a one-semester course on varieties that was designed to "secretly" serve as a launching pad for the followup schemes course in the spring taught by de Jong. The notes, which are the earliest of early drafts for Foundations of Algebraic Geometry, can be found here: http://math.stanford.edu/~vakil/725/course.html. I bring this up because even if you're only looking at varieties right now, you can still consider things from a sheaf-theoretic point of view for fun and profit.

As Michael notes, this will be a good time to start looking at examples. Particularly profitable to learn now might be the theory of divisors on curves, which serves as a model for the more complicated higher-dimensional cases. However, even in the simplest case the theory of divisors is but one facet of something less obvious involving sheaves, their associated geometric line bundles, and transition functions. These objects are all essentially the same, and the better handle you have on the different points of view, the better off you will be. It's a lot to handle, but you have strong background in the right areas.

Solution 4:

To me a first course in Algebraic Geometry should focus on two major things :

Why should one work with schemes ?

• what happens if I try to describe topological spaces/smooth manifolds/etc as locally ringed spaces ;
• for an $\mathbb R$-scheme, what is the difference between $\mathbb R$-points, $\mathbb C$-points and closed points ;
• how $\rm{Spec} \, k[X]/X^2$ can be seen as two points infinitessimaly closed ;
• why is the spectrum defined with prime ideals and not only maximal ideals, what happens in a Jacobson scheme, why are constructible sets so important ;
• why flatness is crucial in Algebraic Geometry, what it means geometrically and how to use it concretly.

Ok, now that I agreed schemes look cool, how does one work with them ?

• many local properties can be checked on complete local rings, so the students should be familiar with the arrows $\mathcal O_X \longrightarrow \mathcal O_p \longrightarrow \widehat{\mathcal O_p}$, in particular with Nakayama ;
• But I would not talk of cohomology, unless they are already familiar with it.