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.
Best Answer
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.