January 5, 2014
June 11, 2013
A new version is now posted at the usual place (the June 11, 2013 version). The main reason for this post is to have a reasonably current version on the web. Thanks to many people for helpful comments — most recently, a number of comments from János Kollár, Jeremy Booher, Shotaro (Macky) Makisumi, Zeyu Guo, Shishir Agrawal, Bjorn Poonen, and Brian Lawrence (as well as many people posting here, whose names I thus needn’t list).
There are a large number of very small improvements, and I’ll list only a few in detail. I’ve replaced the proof of the Fundamental Theorem of Elimination Theory (Theorem 7.4.7). (This new proof is much more memorable for me. It’s also shorter and faster, and generally better. It was the proof I was trying to remember, which I heard in graduate school. I couldn’t find it in any of the standard sources, and reproduced it from memory. But it must be in a standard source, because it is certainly not original with me!) Jeremy Booher’s comments have led to the completion discussion being improved a lot.
I had earlier called , where is the subring of , the “knotted plane”, to suggest the picture that it was a plane, with the origin somehow “knotted” or “pinched”. János Kollár pointed out that “knotted” is misleading, because it is not in any obvious way knotted. I’ve changed this to “crumpled plane”, but this isn’t great either. I’m now thinking about “pinched plane”. Does anyone have a good suggestion for a name for this important example?
Still to do: To repeat my comments from the previous post, as usual, the figures, index, and formatting have not yet been thought about. My to-do list is quite short, so please complain about anything and everything (except figures, index, and formatting). (Perhaps in the next posted version, I may have “TODO” appearing in the text indicating where there is still work in progress.)
Finally, just for fun, here is a picture of the two rulings on the quadric surface, from the Kobe skyline. (It is clear why algebraic geometry is so strong in Japan!)
March 25, 2013
A new version is now posted at the usual place (the Mar. 23, 2013 version). There’s not much to report. I’ve responded to the advice you’ve given in the previous post, done the bibliography (so in particular, you are free to criticize it), given a little more geometric motivation for completions following the advice of Andrei, and responded to more suggestions sent by email, and by in this year’s Stanford reading group.
Still to do: As usual, the figures, index, and formatting have not yet been thought about. (Again: the bibliography is off this list! I’ve gotten advice from a number of people on the index, notably Rob Lazarsfeld, and I have at least some idea of how I want to proceed.) I have a to-do list of precisely 50 items. (Perhaps in the next posted version, I may have “TODO” appearing in the text indicating where there is still work in progress.)
Questions for you: it may soon come time to make figures. Do you have recommendations on how to make pictures? You might guess my criteria: I would like to make them look reasonably nice, but they needn’t be super-fancy; and the program should be easy for me to use (I am a moron about these things), and ideally cheap or free. I’ve used xfig in the past in articles (and the figures currently made), and like it a lot, but I’ve found it imperfect for more complicated figures (with curvy things), and a little primitive for somewhat complicated figures. I’m remotely considering finding someone who is good at this, and seeing how much they might cost.
Added April 26, 2013: Charles Staats sent me this beautiful picture of a blow-up. (I currently haven’t imported it into the file, because for compiling reasons the file goes through dvi, not pdflatex; but don’t bother telling me how to fix this, as I can always ask later if it becomes urgent.) Caution: it didn’t view well on my browser; you may want to download it to view it properly.
February 26, 2013
(Update May 15, 2013: at this point, to my amazement, all of my questions have been answered, although more good answers to the rest would be most welcome. I have learned a great deal of neat stuff I should have known long before by asking these questions!)
A new version is now posted at the usual place (the Feb. 25, 2013 version). There are many small improvements and patches, but no important changes. Because I want to give you something new to look at, here is a newly added diagram of chapter dependencies. I was a little surprised by what it showed.
The “to-do” list of things to be worked on is now at about 100 items. With the exception of formatting, figures, the bibliography, and the index, I continue to be very interested in hearing of any suggestions or corrections you might have, no matter how small. We are clearly nearing the endgame.
I now want to ask advice on a number of issues all at once. These are mostly small things, and are along the lines of “what is the best reference to point learners to on this topic”. Some of the questions are “unimportant”, in the sense that I doubt they will affect my exposition (although they may be important in some larger sense). For those questions relating to some particular part of the notes, I will give the section number. Please feel free to respond by email or in the comments. Here we go!
Peter Johnson strongly preferred using “fibre” instead of “fiber”. Does anyone else feel strongly?
1.4.1 Latex question: how do you get the \varprojlim subscript in the right place? (answer here)
1.6.12 (unimportant) Do right-exact functors always commute with colimits? (For example, commutes with direct sums, which is what we use, but that is easy to check directly.) (answer here)
5.4.M Can anyone get this exercise (that, basically, says that normality descends under finite field extensions)? I think it should be gettable, but not easy, but I’ve had clues that this is harder than I thought. I want to be sure I have the level correctly gauged. (Feel free to respond by email if you get stuck.) (two positive response received so far, including this one)
5.4.N is not a unique factorization domain, but its Spec can be covered with 2 (distinguished) affine subsets, each of which are Specs of UFD’s. Is there some good reference for this? (Presumably it becomes a UFD upon inverting either 2 or 3, but I can’t see why this is the case. And of course I don’t just want to know what is true; I’d like a reference for why it is true.) Added later: I should also have added, is there a well-loved reference that shows that the class group of is ? (answer here)
6.3.K A compact complex variety can have only one algebraic structure. What is a reference? (A number of sources mention this fact, but I want an actual proof.) On a related point, in 10.3: A variety over is proper if and only if it is compact in the “usual” topology. What is a reference? (answer here and here)
6.7 In this section, I mention the Schubert cell decomposition of the Grassmannian. The key idea is that any -dimensional subspace of (where is a field; and say , …, is the standard basis of ) has a canonical basis, where the first to appear in each basis element appears with coefficient 1, and that appears in no other basis element, and that special for that basis element is “to the right” of the of the previous one. Is there a standard name for this? (Normal form? Row-reduced echelon form?) Is there a good (fairly standard) reference for it? (Perhaps this gets too far into how linear algebra is taught in different countries, and I should just not give a reference, and instead give it as an exercise.) (answer here, although I’m also happy to get more references)
8.4.H Interesting fact: I almost wanted to say that effective Cartier divisors are the same as codimension 1 regular embeddings. But I could only show this in the locally Noetherian situation (or more generally, when the structure sheaf is coherent). The reason for the problem is that the definition of effective Cartier divisor is in terms of open subsets (for good reason), while the definition of regular embedding is in terms of stalks (for good reason), and getting from the latter to the former requires Nakayama. If you think I’m not giving the right definition of one of these two notions, please let me know! (see here for an interesting follow-up, thanks to Laurent Moret-Bailly)
9.1.7 Peter Johnson did not like my use of the phrase “open subfunctor” in 9.1.7. Is anyone else bothered? How seriously? (current plan after discussing with Peter: leave as is)
9.4.E Can anyone get this exercise (that, basically, says the product of integral varieties over an algebraically closed field is also an integral variety)? I think it should be gettable, but no one I know has gotten it (possibly because I haven’t asked it in homework sets). I want to be sure I have the level correctly gauged. (Feel free to respond by email if you get stuck.) (two positive response received so far, including from Gyujin Oh)
10.3.9 Is there an example of a non-smooth group variety over a field , i.e. a finite type reduced group scheme over that is not smooth? Translation: is there a group variety that is not an algebraic group? (answer: yes! example here)
11.3.13 Over an algebraically closed field, every smooth hypersurface of degree at least in is not uniruled. What is a good reference? (I know why it is true! As with many of these questions, I’d like to know where to point people to.) (answer here)
13.8 I mention Tate’s theory of non-archimedean analytic geometry. Is there a “right” source to point the interested reader (who is just starting out) to? (possible answer here)
19.9.B In (126.96.36.199), we have , and the discussion is away from characteristic . I want to say that the normalization factor is because of characteristic , but I couldn’t convince myself that this was true. Presumably it is. Is there a good reference? (Remark for comparison: one can also write in terms of : . Here the prime factors of are and ; but the reason for the is not characteristic .) (answer: yes, see here)
20.2.H Suppose is a complex elliptic curve. Then is always or . It is if there is a nontrivial endomorphism from to itself (i.e. not just multiplication by followed by translation); the additional class comes from the graph of this endomorphism. Is there a reference for this fact that I can/should direct learners to? (answer: yes, see here)
21.5.9 Is there a good reference for the Lefschetz principle? (Examples currently mentioned: Kodaira vanishing in characteristic 0; and non-jumping of hodge numbers in characteristic 0.) (good answer here)
21.7.8 (not needed) It is a nontrivial fact that irreducible smooth projective curves of
genus have finite automorphism groups. I know three arguments: using the Neron-Severi theorem (and the Hodge index theorem) (see Hartshorne V.1.9, for example); the fact that the automorphisms are reduced and form a scheme (too hard); and by action on Weierstrass points. I am surprised that this is so hard. (Note: I know that the idea can be quickly outlined to someone learning. But I want an easy complete rigorous proof. As long as I am asking, I also want someone to give me a Tesla Roadster.)
21.7.9 Smooth curves in positive characteristic can have way more than automorphisms. Is there a “best” reference?
28.1.L Is there a canonical reference for Tsen’s theorem, that any proper flat morphism to a curve, whose geometric fibers are isomorphic to is a Zariski -bundle? Follow-up question (posted March 5), in response to David Speyer’s comment here: Does anyone have a (loved) reference for the fact that the universal plane conic (over the space of smooth plane conics) is not a -bundle? (See David Speyer’s comment for a little more detail.) (possible answer to the first question here; answer to the second question here)
29.3.B I currently define node only in the case of a variety over an algebraically closed field, in which case I say that it is something formally isomorphic to . I gesture toward the definition in other cases. For example, if is not algebraically closed, I define it as , where is a quadratic with no repeated roots. I want to say that if is reducible, then this is said to be a split node, and otherwise it is a non-split node. I’d thought this was standard notation, but google suggests otherwise. Does anyone have strong feelings about this?
29.5 (This is a follow-up to discussion in the 27th post.) I am reluctant to introduce new terminology in a well-established field, but there is a notion that I think deserves a name. Suppose is a proper morphism. (For the technically-minded, it is likely that “finitely presented” should also be added, but I will play it safe, and not include this.) Then I want to say that is [something] if the natural map is an isomorphism. Not EinStein suggested the name -connected, and I quite like this — it suggests that this notion is even stronger than connected, and suggests in what way it is stronger. Another possibility is -isomorphic (which I suggested, but which I currently like less well). Opinions? (Are you offended by giving this a new name? Or do you like one of these suggestions? Or do you have another idea?)
30.3.4 Is there a canonical (“introductory”) reference for (which will require an introduction to derived categories)? (Brian Conrad’s book Grothendieck duality and base change perhaps?) (possible answer here)
February 19, 2013
The twenty-seventh post is the February 19, 2013 version in the usual place.
Drafts of the final two chapters are now complete. At this point, all the mathematical material is essentially done. The list of things to be worked on is now strongly finite (well under 150 items). With the exception of formatting, figures, the bibliography, and the index, I am very interested in hearing of any suggestions or corrections you might have, no matter how small.
Here are the significant changes from the earlier version, in order.
The first new chapter added is the Preface. There is no mathematical content here, but I’d appreciate your comments on it. These notes are perhaps a little unusual, and I want the preface to get across the precise mission they are trying to accomplish, without spending too much time, and without sounding grandiose. I’ve noticed that people who have used the notes understand well what they try to accomplish, but those who haven’t seen them are sometimes mystified.
In 10.3.9 there is a new short section on group varieties, and in particular abelian varieties are defined, and the rigidity lemma is proved. Although it isn’t possible to give an interesting example of an abelian variety in these notes other than an elliptic curve, it seemed sensible to at least give a definition.
In 20.2.6, a short proof of the Hodge Index Theorem is given (on the convincing advice of Christian Liedtke).
And the last new chapter is Chapter 29, on completions.
29.1 is a short introduction.
29.2 gives brief algebraic background. It concludes with one of the two tricky parts of the chapter, a theorem relating completion with exactness (and flatness).
In 29.3 we finally define various sorts of singularities.
In 29.4, the Theorem of Formal Functions is stated; this is the key result of the chapter. Note: the proof is hard (and deferred to the last section of the chapter). But other than that, the rest of the chapter is surprisingly (to me) straightforward.
In 29.5, Zariski’s Connectedness Lemma and Stein Factorization are proved. As a sample application, we show that you can resolve curve singularities by blowing up.
In 29.6, Zariski’s Main Theorem is proved, and some applications are given. For example, we finally show that a morphism of locally Noetherian schemes is finite iff it is affine and proper iff it is proper and quasifinite.
In 29.7, we prove Castelnuovo’s Criterion (paying off a debt from the chapter on 27 lines), and discuss elementary transformations of ruled surfaces, and minimal surfaces.
Finally, in 29.8, the Theorem of Formal Functions is proved. I am following Brian Conrad‘s excellent explanation (and I thank him for this, as well as for a whole lot more). It applies in the proper setting (not just projective), and is surprisingly comprehensible; it builds on a number of themes we’ve seen before (including Artin-Rees, and graded modules). I think Brian told me that he was explaining Serre’s argument. The proof is double-starred, but I hope someone tries to read it, and makes sure that I have not mutilated Brian’s exposition.
I’m going to continue to work on the many loose ends, and to fix things that people continue to catch. There are also a number of issues on which I want to get advice (on references, notation, etc.). I think it makes sense to ask all at once, rather than having the questions come out in dribs and drabs (as in that case people may read the first few, but then stop paying attention). So I intend to do this in the next post, and likely within a month.
Here is an example of the sort of thing I will ask, that is relevant for the chapter just released. There is a kind of morphism that comes up a lot, and thus deserves a name. Suppose is a proper morphism of locally Noetherian schemes (Noetherian hypotheses just for safety), such that the natural map is an isomorphism. Can anyone think of a great name for such a morphism? I’d initially thought about using “Stein morphism”, but that’s terrible (as pointed out by Sándor Kovács), because it suggests something else (from complex geometry). Sándor has suggested “connected morphism”, and Burt Totaro correctly points out that this is the “right” version of “connected fibers”, but this seems imperfect because it suggests something slightly wrong. I think that “contraction” would be good, but that’s already used in higher-dimensional geometry (more precisely, the contractions there are these types of morphisms). Someone (my apologies, I can’t remember who) suggested “algebraic contraction”, which seems somehow better. But for now, I’ve not called it anything, and perhaps it is better that way.
December 17, 2012
The twenty-sixth post is the December 17, 2012 version in the usual place. A large number of small improvements have been made, and the exposition has converged substantially, although there isn’t much big to report.
The (small) changes:
The terminology “local complete intersection” is changed to “regular embedding” (the more usual language, along with “regular immersion” — note that I have gone with “embedding” rather than “immersion” throughout). This was because the notation I was using would cause confusion because of the existence of an importance class of morphisms called “local complete intersection morphisms” (“lci morphisms”).
I am more careful about distinguishing the canonical bundle of a smooth projective variety (the determinant of the cotangent bundle) from the dualizing sheaf, before they are identified in the last chapter, to avoid anyone being confused about what is being invoked when.
A proof of the classification of vector bundles on the projective line (sometimes known as Grothendieck’s Theorem) is given in 19.5.5. (A proof is possible by painful algebra, and another proof is possible using Ext’s. Because of what else is discussed in the notes, I take a middle road: an easy proof without Ext’s, which relies on an easy calculation with 2 by 2 matrices. Note: I have not done the important fact that classifies extensions, which has tied my hands a little.) This required popping out “a first glimpse of Serre duality” into its own section (19.5).
Some discussion relating to Poncelet’s Porism is added in 20.10.7.
Still to do.
The introduction and the Zariski’s Main Theorem / Formal Functions chapter are still to be written. (And of course things like the index, figures, and formatting won’t be dealt with until the end.) The only other substantive things still to be written are a brief discussion of radicial morphisms (for arithmetic folks) and a brief proof and discussion of the Hodge Index Theorem (for geometric folks). Other than that, I expect essentially no other new material to be added.
I’ve caught up with almost all of the corrections and suggestions people have given me, except for a double-digit number of pages from both Peter Johnson and Jason Ferguson. (Thanks in particular to the comments of the reading group at Stanford for useful comments: Macky, Brian, Zeb, Michael, Evan, and Lynnelle!)
My to-do list still has 212 things on it. But that is a drastic improvement; the notes are converging rapidly.
I will not make any further progress until January. I hope to have a draft of the introduction done in January, and the final substantive chapter in February.
Happy holidays everyone!
October 10, 2012
The twenty-fifth post is the October 10, 2012 version in the usual place. (Update Oct. 24: a newer version, dated October 23, 2012, is posted there now. Some of the changes are discussed in the fourth comment below.) The discussion of smooth, etale, and unramified morphisms has been moved around significantly. Johan de Jong pointed out that “unramified” should best have “locally finite type” hypotheses, thereby making its link with the other two notions more tenuous; and Peter Johnson pointed out that one could give the definition of smoothness much earlier, at the cost of initially giving an imperfect definition (a trade-off I will happily take).
I am very interested in having these changes field-tested. (Most of the rest of the notes are now quite robust thanks to the intense scrutiny they have been subjected to.) I know that when something is revised, the revisions are looked at much less. But I am hoping that someone hoping to learn about smoothness, or solidify their understanding, will give this a shot in the next couple of months. I know that in the course of doing this, my understanding of these ideas has been radically improved. Because the actual algebra was elided in most of the “standard sources”, I hadn’t realized what was important and what was unimportant, and what didn’t need to be hard and what needed to be hard. So I can at least make a promise to many readers that they might learn something new.
Here are the changes, along with suggestions of what to read (for those who have read earlier versions).
Chapter 13: Nonsingularity
13.2.8 The Smoothness-Nonsingularity Theorem is an important player. (a) If k is perfect, every nonsingular finite type k-scheme is smooth. (b) Every smooth k-scheme is nonsingular. This gets stated early, but proved late. To read: the statement of the Theorem. (To experts: Am I missing easy proofs? I think it has to be as hard as it is. Update Oct. 24, 2012: David Speyer and Peter Johnson have outlined proofs in the comments below, using just the technology of Chapter 13.)
In 13.4, I had a bad exercise, which stated that if l/k is a field extension, and X is a finite type k-scheme, then X is smooth if and only if its base change to l is smooth. One direction is easy, but I’m not even sure how to do the other direction at this point in the notes. This converse direction is now 22.2.W, which I’ll discuss bellow. To read: nothing.
13.7 is the new section on smooth morphisms, including a little motivation. Everything is easy, except showing that this definition of smooth morphisms correctly specializes to the older definition of smoothness over a field. (Notational clash that I have not resolved: the “relative dimension” of a smooth morphism is n in this section, but was d earlier. There are reasons why I couldn’t change the n to d and vice versa. I don’t think this will be confusing. (But in general I have tried hard to be consistent with notation.) To read: these 3 1/2 pages.
Chapter 22: Differentials
22.2.28-30 (a very short bit): Here a second (third?) definition of smoothness over a field is given (as we can now discuss differentials) — this was in the older version. The second definition allows us to check smoothness on any open cover, for the first time, which in turn allows us to more easily check (in 22.2.W) that smoothness of a finite type k-scheme is equivalent to smoothness after any given base field extension. This in turn allows us to establish an important fact in 22.2.X: a variety over a perfect field is smooth if and only if it is nonsingular at its closed points. This had early been in Chapter 13, but relied on 22.2.W. This also establishes part of the Smoothness-Nonsingularity Comparison Theorem. To read: 22.2.W and X (very short). Update Oct. 24, 2012: in the Oct. 23 version, this is now made into a new section, 22.3, which also includes generic smoothness. 22.2.W and X are now 22.3.C and D.
22.5: Unramified morphisms are now discussed here. This section is easy. To read: 1.5 pages. (Update October 24, 2012: the new section 22.3 bounces this section forward to 22.6 in the Oct. 23 version.)
Chapter 26: Smooth, etale, and unramified morphisms revisited
This chapter is notably shrunk. 26.1 still has motivation, but now the definitions I used to give are now just “Desired Alternate Definitions”. 26.2 now discusses “Different characterizations of smooth and etale morphisms, and their consequences”. The central (hard) result is Theorem 26.2.2, which gives a bunch of equivalent characterizations of smoothness. Before proving it, a number of applications are given. The statement of Theorem 26.2.2 is rearranged, the proof is the same. To read: skim 26.1 and read 26.2 up until the (and not including) the proof of 26.2.2: 4 light pages.
There are some problems I would still like to see worked out by real people.
26.2.E: I moved this from the “unramified” section, but should probably move it back, as I think it can be done with what people know there.
26.2.F: Is this gettable?