The eighteenth version of the notes is the September 6 version in the usual place. The most important advance since the seventeenth version are a number of changes in response to many suggestions from various people (most recently some insightful ones from Jakub Byszewski). A chapter on the 27 lines is also added — more on this below. There will a pause in the posts for this reason:
I owe responses to a number of recent comments and emails.
Here is where the project currently stands. The year-long plan of putting most of the notes online is now declared complete, and is more complete than I’d hoped at the start. (Only) three substantive topics (chapters) remain to be added: smooth/etale/unramified; regular sequences; and formal functions (and related topics: Zariski’s main theorem, Stein factorization, etc.). A reasonable amount of topics need substantial editing in response to comments, and there are some small topics I still want to add. Other things will be added much later, including a proper introduction, bibliography, and index.
This academic year, I will teach a three-quarter graduate class based on these notes, so my focus will shift to where in the notes the class is. I will test my assertion that the important parts of this material can be covered in a year-long class that will be considered a success by a reasonable number of those students who last until the end, and that a reasonable number of students will last until the end. (I also want the course to be considered a success by students who choose to take only the first quarter, or the first two. It should be considered worthwhile — even if hard work — for students not intending to become hard-core algebraic geometers, and there should be a nontrivial number of these.) The webpage for the class will be on this “blog”, and I will welcome emails and comments from people whether or not they are enrolled in the class.
In a course like this, it is essential that students solve a large number of problems, and get feedback on them. Unfortunately, we don’t have the resources for a grader, so I’m trying to think of some creative alternative. In quarters two and three of previous incarnations of this course, I’ve had people hand in problem sets, and have kept track of what problems people did, and read some of their solutions. It was far from ideal, but was better than nothing. This will be tricky for me, especially at the start of quarter one, as there will be more students (and I need to quickly convince those without sufficient background to take other classes instead), and the baby will be very small (and time and attention and sleep will be in short supply). One possible alternative is to require enrolled students to read and grade each others’ homework — say they have one week, during which they have to grade three of the problems that I pick. Students would hand in homework to me electronically (e.g. scanned, if their solutions were handwritten). I would glance at the solutions, keep track of who did what, pick the problems in each set to be graded, and then divide up the submissions among the enrolled students (so even unenroled students would get feedback), and give them a week. This would be a bit slow and cumbersome. Could this work? Has anyone tried this, or anything else (besides just letting students work on their own with no feedback — something which is standard, but which I think is far from ideal)?
I’ve redone Chevalley and related things, including the proof that codimension is the difference of dimension for varieties. Any comments on this (or anything else) would be appreciated.
There is currently no definition of generically finite in the notes. This is because it isn’t clear to me what the accepted definition is, even though this phrase is tossed around in talks. I was pleased to recently find out that Johan de Jong was in a similar position; the section of “generically finite morphisms” in the stacks project had some theorems, but no definitions. We had a discussion on what the definition is, and the result is here. What do you think of these two definitions? Or is there some official definition in the literature?
I remain confused on the right definition of Hilbert function. It is one of two things. (i) If M is a finitely generated module over a graded ring, it can be the dimension of the various graded pieces. Special case: the graded ring of a projective variety. (ii) If X is a projective variety (with embedding into projective space), it can be the dimension of the restriction of degree n polynomials (in the projective coordinates) to X. I want to use (i), but fear that (ii) may be right, at least for some people. (The definitions disagree: consider 3 distinct points on a line, where the value of the first polynomial is 3, and the value of the second polynomial is 2.) Can anyone give an opinion, informed or otherwise? (This was discussed earlier here.)
The twenty-seven lines
I’ve thought through the 27 lines, and have tentatively decided that this is a worthwhile fun chapter to have near the end of the notes. I am tentatively hoping to end the 2011-12 course with this topic. It is beautiful, and also connects a number of ideas and themes. (On the other hand, I’m trying to hew to a tough line about what gets included, so I’m seriously considering removing it again.) As usual, comments would be very appreciated — if you’ve always wanted to learn why there are 27 lines on every cubic surface (not every standard source has a complete proof, although this sometimes is not clear), and want something readable, please take a look (and when you find it isn’t readable, complain). There are a few references to the chapters not yet public (Castelnuovo’s criterion; miracle flatness); please excuse them, and take them as black boxes.
After thinking this through, I’ve had some thoughts on this question, which I may as well record here. There are a number of different possible results one can prove. (a) One can show that every smooth cubic (over an algebraically closed field) has precisely 27 lines. (b) One can show that blown up at six points (in suitably general position) can be anticanonically embedded as a cubic surface, and such surfaces have precisely 27 lines. (c) One can show that every smooth cubic is a blow-up of , and hence use (b) to prove (a). Different people do different things. It is possible to prove (a) rather quickly and in a low-tech manner. (Miles Reid’s Undergraduate Algebraic Geometry does a great job of this. I am a Miles Reid fan in general.) One can show (b) relative quickly, although the “embedding” part can be a bit annoying. (Hartshorne follows this route.) With Castelnuovo’s criterion (which requires formal functions), one can show (c) (and hence (b) and (a)); to get this going, you need two skew lines in the cubic surface, and precisely five lines in the surface meeting them both.
I decided to do something slightly longer, and prove (a) first, by showing the key result that the space of lines in a fixed smooth cubic surface is reduced of dimension 0. I find this enlightening for a number of reasons I won’t spell out here, even though it comes down to an explicit calculation. I’d be interested to hear what opinions people have on this; I may be convinced to save some space, and just directly find the two skew lines and five lines connecting them.