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My hope of a new version each month has clearly not worked out! But here is one for April, just before the month ends. As usual, a random selection of corrections and minor improvements have been dealt with, and many more remain to do.

The April version is posted at the usual place (the April 29, 2015 version).

I apologize in advance for all of the following.

  • I won’t have a chance to follow comments as frequently as I would like to. At this point, I expect to check in roughly once per week, with some big gaps during busy periods (conferences; fall recommendation periods; family illnesses; etc.).
  • I hope to be able to respond to most comments deserving a response, but I fear I won’t be able to.
  • If you would like to respond to something privately, feel free to email me. As others who have tried to email can tell you, I get enough email that I’m unable to respond to more than a fraction (and even that not in a timely manner), but I certainly try.
  • Posts will be retroactively edited to fix problems.
  • The many comments people make will lead to improvements of the notes (or at least changes!). I apologize in advance for not properly acknowledging everyone individually for their help.

Having gotten all of that off my chest, I feel much better. I hope you do too!

I’m hoping people will take a look and make comments.  They broadly fit into three groups.

  • Perhaps a few brave students (with too much time on their hands, and serious background) will attempt to treat this as a world-wide reading course.  (Warning:  this will be very hard, and a lot of work.   You will have to do the homework.)  If you are hoping to get credit for this, you should set things up with a supervisor at your home institution.  What problems can you do?  What problems are too hard? What explanations are harder than necessary?  (You may not be able to evaluate the “than necessary” part.)
  • Perhaps people who have recently learned schemes and are solidifying their knowledge will dip and and out.  The same questions as above apply.
  • Perhaps curious and kind-hearted experts will take a look and make suggestions and corrections.  (There are certain experts in particular I hope will drop in.)  In particular, are there key examples or simple explanations or important links missing?  The only thing experts are less useful for: judging how hard topics are. I find it too easy to forget what is hard the first time through (e.g. the correspondence between line bundles and divisors).

Feedback I’d like:

  • “I found these errors: …”
  • “I found these typos: …”
  • “Topic X should certainly be learned in a first year of algebraic geometry. Why didn’t you include it?”
  • “Here is a great explanation of this theorem.”
  • “Your explanation of this idea was confusing than it needed to be.” (And possibly: “Here is a much better [or different] way of explaning that.”)
  • “Here is a cool example I wish someone had told me when I was younger.”
  • “My students [or I] had a hard time with this notion.”

I fully appreciate that the students I’ve had are not typical, and that these notes are not suitable for most people.

Comments still to respond to: I’ve dealt with all 4.

Although this uses wordpress technology, this is not a blog. The purpose of this site is to get advice and suggestions on continually evolving notes on the foundations of algebraic geometry.

These notes deal with schemes, and attempt to be faster and more complete and rigorous than most, but with enough examples and calculations to help develop intuition for the machinery. Such a course is normally a “second course” in algebraic geometry, and in an ideal world, people would learn this material over many years, after taking serious courses in commutative algebra, topology, complex analysis, differential geometry, homological algebra, and number theory.  We do not live in an ideal world.

I’ve officially taught this course at Stanford three times (and unofficially ran reading courses more often than that), and my lecture notes have been converging over time.  The notes distill things people have told me through the years, of slick explanations of the basic things one needs to know to work in algebraic geometry.  I would now like to refine them further, by digesting into it more collective wisdom.

My intent for the 2010-2011 academic year (September 2010 – August 2011) was to gradually and sequentially edit them, and to post them, roughly at the rate of a (hard, fast) course.  This goal is now complete. My intent for the 2011-2012 academic year is to teach a year-long graduate course at Stanford, and continue to post the notes, and continue to have interesting discussions with people via this site, which will lead to ongoing revision and improvements.

See these mathoverflow answers for some interesting discussion just before this started.