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Small post: new version

Posted by ravivakil under

Actual notes
[33] Comments
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.

March 25, 2013 at 5:03 pm

Pictures are always a pain. I have to draw a lot of them, and I’m a big fan of Inkscape (see http://inkscape.org/). Perhaps the most annoying thing is labeling them (you really want TeX to typeset your labels). For this, I use the pinlabel package that Colin Rourke wrote (it’s also used in eg G&T and AGT when they typeset papers). For this, it is helpful to use either Pete Storm’s pinlabeler (see http://www.math.upenn.edu/~pstorm/pinlabeler.html) or Nathan Dunfield’s labelpin (see http://www.math.uiuc.edu/~nmd/software/).

My wife used to be an architect and does great technical drawings; however, I’ve found that it takes more time for me to describe what I want to her than to just draw things myself. So be warned that hiring someone might not be as efficient as you think.

March 29, 2013 at 5:49 pm

Thanks Andy! I agree that “if you want something done the way you want, you should do it yourself” (even if it is less than perfect), so I will undoubtedly be doing these myself.

April 16, 2013 at 10:04 am

As you have mentioned Inkscape and below there is a discussion about TikZ (a recommendation I’d like to second): There is actually a plugin for Inkscape to export graphics to TikZ code, so they can be directly embedded into Latex documents:

http://code.google.com/p/inkscape2tikz/

Making pictures hard-coded in TikZ can be fairly time-consuming, so this might help save some time. Especially when the results don’t have to be “super-fancy”.

April 17, 2013 at 1:38 pm

Dear Gregor,

Thanks! This looks very useful! (Hopefully others will take advantage of this advice as well.) best,

Ravi

April 22, 2013 at 12:56 pm

For two-dimensional pictures, Inkscape is great. I use it all the time.

As for the labels, there is a very good and easy solution. Inkscape lets you export a picture to pdf. In the dialog that appears, you can select an option called “PDF+LaTeX”.

If this is enabled, Inkscape will make a pdf of your picture, but leave out all the text. It will also create a tex-file with commands to include the picture and to put all text in the picture in the correct place.

You now have to include the Inkscape-generated tex-file using \input (where you would normally use \includegraphics). The text will be selectable in the resulting pdf and will use LaTeX’s font and size. Including math or even \ref’s is not a problem. Absolutely great.

More info at ftp://ftp.dante.de/tex-archive/info/svg-inkscape/InkscapePDFLaTeX.pdf

April 26, 2013 at 12:36 pm

Daan, that is very slick — thanks for letting me know!

March 27, 2013 at 8:07 am

Among the tools to draw commutative diagrams as surveyed by James Milne (http://www.jmilne.org/not/CDGuide.html). TikZ is my favorite. One reason is the pictures produced by TikZ are really beautiful. The second reason is that TiKZ codes are well integrated with LaTeX codes. The last reason is that TiKZ can be used to create more complicated pictures which other commutative diagram packages cannot do.

March 29, 2013 at 5:49 pm

Thank you Fei! I’m looking forward to checking this out.

April 10, 2013 at 11:10 am

Follow-up:

Greg Smithalso recommends TikZ.Here is a linkto some sample pictures made by it. Greg also says that putting latex in figures is essentially trivial (a point implicitly made by Fei above). This is pretty convincing!April 13, 2013 at 11:07 am

Another note: the tikz-cd package uses a syntax very similar to that of xy-pic to produce commutative diagrams with tikz. In my opinion, the diagrams so produced are significantly better-looking than those produced with xy-pic, especially when viewed electronically.

April 14, 2013 at 4:34 am

Another good reason to go with TikZ!

March 29, 2013 at 5:52 pm

David Savitttold me thatDanny Calegarihad written a software package for making pictures, and Danny confirmed that this is true: it is called wireframe, and with it, it is very easy to make a figure, and to export it as an .eps file for inclusion in a LaTeX paper. The main point of the program is to draw nicely rendered pictures of three-dimensional objects, especially surfaces (obtained e.g. by taking the boundary of a thickened neighborhood of a graph, and then smoothed by subdivision and approximate curvature flow).You can read about ithere.March 29, 2013 at 5:55 pm

I also want to point out that Mehdi Omidali has made some beautiful figures to accompany the notes, which I mentioned

here. At some point I will figure out (from his earlier emails to me) how he did such wondrous things, and let you all know.April 15, 2013 at 6:23 am

Here is a question for you.

Suppose is a morphism of irreducible varieties of the same dimension. Then there is a dense open subset of over which is finite. (I think of this as “generically finite is the same as generally finite”.) It seems a nice thing to know. My current proof involves a hard fact, and I wanted to check if I am missing anything.

The discussion in section 11.4 on “Dimensions of fibers of morphisms of varieties” almost proves this fact. (You needn’t look at this section in order to read or answer this question.) In particular, the proof of Proposition 11.4.1 almost works. We first reduce to the case where is affine. Then if is also affine, the argument of Proposition 11.4.1 essentially applies.

What then to do if isn’t affine? Then obviously you cover it with finitely many affine open subsets , …, . For each of these, you can shrink further so that above this open subset, every is finite. How do you know that the union is finite?

My best current argument is that proper + quasifinite = finite (Theorem 29.6.20, which uses Zariski’s Main Theorem). The finite

separatedunion of proper things is proper, and the finite union of quasifinite things is quasifinite, so we win.Is there some easy argument that just shows that if we have a separated morphism , and has a finite cover by , …, such that is finite for each , then is also finite?

April 17, 2013 at 3:24 am

Still no replies? It’s not deep, just a matter of repeated shrinking.

Given \pi: X -> Y between irred. k-varieties of same dim. n, must also

assume dominant. Function-field ideas give affine opens on which \pi

induces an isomorphism U –> V. Want to control inverse image of V.

Can shrink Y to V. Choose a finite cover of X by affine opens X_i.

All contain the generic point of X. Shrink U, V to get U in all X_i.

Suffices to show V (affine) has a nonempty open where fibers of \pi_i:

X_i -> V are singletons. Sketch: rings on both sides are f.g. over R =

k[x_1, .., x_n]. Any new generator satisfies a nonzero equation over R.

Look at coefficients to define some f in R for which points (primes) in

the subset D(f) of Spec R lift uniquely. Do on both sides, and shrink.

April 17, 2013 at 6:31 am

Well, as so often my first try was on the right track but with incorrect details. So here is a fully revised post:

Asssume \pi: X -> Y is a dominant morphism between irred. k-varieties of the same dimension n. Shrink Y to make it affine. Choose a finite cover of X by affine opens X_i and an affine open U contained in their intersection. As U contains the generic point of X, its image is dense, hence contains a nonempty open subset V’ of Y.

It will be shown below that Y also has nonempty opens V_i where fibers of \pi_i: X_i -> Y are singletons. One then uses an affine open V in the intersection of these V_i and V’. The construction gives a sheaf morphism \pi^{-1}(V) -> V that is a homeomorphism, with domain contained in U. Using some sort of affine communication, can shrink this to get a finite morphism between affine opens, the desired result.

Sketch for the above: rings on both sides of X_i -> Y are f.g. over R = k[x_1, .., x_n]. Any new generator y satisfies a nontrivial polynomial relation r_y.y^N + lower terms, N minimal, over R. The product of the r_y, for enough y to generate the larger ring, is an f in R for which points (primes) p in the subset D(f) of Spec R lift uniquely. To see this, it helps to localize the larger ring at the subset R-p, still an integral domain, whose quotient by p is a finite extension of QF(R/p), hence a field. At the topological level, the bijection is clearly a homeomorphism.

April 17, 2013 at 1:43 pm

Hi Peter,

This is an interesting approach, I don’t fully see the argument yet. In particular, I’m nervous about the assertion that “Y also has nonempty opens V_i where fibers of are singletons”. If I understand it correctly, this isn’t true of the double cover corresponding to given by (in characteristic not 2).

April 17, 2013 at 4:56 pm

Hi Ravi,

You are right; I was too hasty. That part of my plan was too much to expect. But doesn’t something more need to be known about inverse images, to check the “affine” property finite morphisms must by definition have? At the moment (lacking time to think properly) I don’t see that this part has yet been resolved.

April 17, 2013 at 1:39 pm

Daniel Litthas given me a fast argument, which you can seehere. Thank you Daniel!(Addendum, Apr. 19, 2013: the linked version is one day newer, as I realized I could make the argument slightly cleaner.)April 17, 2013 at 6:35 pm

OK, now I see. The generic point of Y has 0-dimensional fibers, so has a unique inverse image, the generic point of X. The complement of any affine open U of X is the closure of a finite number of points, and the closure of their image in Y avoids the generic point. This gives a nonempty affine open in Y with inverse image contained in U, thus reducing to the solved case of affine schemes.

April 29, 2013 at 9:09 pm

Dear Ravi,

Minor typo report: p.14, para. 3 of the March 23 draft, “There many other …” is missing the word “are”.

Cheers,

Matt

April 30, 2013 at 6:34 am

Thanks Matt — now fixed!

May 29, 2013 at 4:57 am

Stefan Kebekusjust showed meQTikZ, where it compiles TikZ in real time, so you can muck about with parameters until it looks right.June 6, 2013 at 7:43 pm

I have a typo and two questions in the version 2013/3/23.

(1) p279,line -9: subscheme $X$ SHOULD BE subscheme of $X$

(2) p279,line -8: is a variety itself.: I verified the separatedness and finite-typeness, but the reducedness is also true under your definitions of varieties and subvarieties?

(3) P398,line 4,5: is a unique factorization domain, class group $0$.: I think they are false. For example $z^2=(x+iy)(x-iy)$ in $C[x,y,z]/(x^2+y^2-z^2)$ or $k[x,y,z]/(xy-z^2)$ is not a UFD.

I would be sorry if I am misunderstanding something.

June 7, 2013 at 10:38 am

Hi Frank,

You are completely right on all counts, and I’ve fixed them all (in the version that should hopefully be posted within a week). (About (2): reducedness hypotheses are necessary, and now added.) You are a very careful reader! I’ve now fixed all of these.

June 7, 2013 at 5:17 pm

Hi Ravi,

Thank you for quick response. The following are additional possible typos. All are minor though I might be misunderstanding something in a few of them.

—–

p65:Figure 1.1:Clearly $S^{0,k}$ SHOULD BE Clearly $S^{k,0}$

p66:1.7.H:line -2: In particular, $S_\infty^{0,k}$ SHOULD BE In particular, $S_\infty^{k,0}$

p66:1.7.11:line 2:$Y_r^{p,q}:=d(S+S)/S$ SHOULD BE $Y_r^{p,q}:=(d(S)+S)/S$. (subscripts of S omitted)

p109,3.2.P(b):last line: $f_n(a_1,…,a_n))$ SHOULD BE $f_n(a_1,…,a_m))$

p170:5.5.N(b),line 1:$p\in A$ SHOULD BE $p\in Spec A$

p206:7.3.C(b):line2: is cover SHOULD BE is a cover

p211:7.3.M,line4: inducing finite SHOULD BE including integral

p213,7.3.R(d): integral over Spec SHOULD BE integral over

p241,8.4.G,line 6: such that in $O_{X,p}$ SHOULD_ BE such that in $O_{Y,p}$

p249,line 6:$h_{X\times_Z Y}$ SHOULD_ BE $h_X\times_{h_Z}h_Y$

p281,line6: $\delta_X$ SHOULD BE $\delta_W$

p285,line 5: $s$ need not … if $\rho$ is not SHOULD BE $\sigma$ need not … if $\mu$ is no

p288,middle picture: $\Gamma_f$ SHOULD BE $\Gamma_\pi$

p305,11.2.F,line 3: is the SHOULD BE is a

p307,line 11: $\in A$ is SHOULD BE $\in B$

p307,line 12:

$Q_1\cup S$ SHOULD BE $Q_1\cup B$

p307,line 12:also in $A$ SHOULD BE also in $B$

p307,line 13:$P\cap A$ SHOULD BE $P\cap B$

P330,line1: only the SHOULD BE only if the

p343,line -17: any element of $K(A)$ SHOULD BE any element of $K(A)^x$

p355,line–6: generated $A_\dot$-module SHOULD BE generated $A_\dot(I)$-module

P362,13.1.H,line2: of ran SHOULD BE of rank

p362,line -9:apply Exercise 13.1.I(??)

P381,line2: stalk ${\cal F}|_q$ SHOULD BE talk ${\cal F}|_q$

—–Fin

June 7, 2013 at 5:22 pm

P381,line2: stalk ${\cal F}|_q$ SHOULD BE talk ${\cal F}_q$

June 10, 2013 at 11:11 am

Thanks! I’ve gone through all of these. You are right on all counts. A few had been caught earlier, but the vast majority hadn’t, so please let me know if you have any more like these. The self-referential issue on p. 362 is still to be fixed, but on my to-do list.

June 10, 2013 at 11:22 am

Thanks Frank! A response is now posted.

June 11, 2013 at 2:36 am

Thanks Ravi,

>please let me know if you have any more like these.

I am happy to increase readability even slightly for the new standard textbook.

It would be appreciated if you will let me know when I am wrong in the list of typos.

—

Typos in (ver2013.3.23):

p119,3.6.M,line 4: $(x-a,y-a)^2$ SHOULD-BE $(x-a,y-a^2)$

p234,Example2,line2: given by $k[x]/x$ SHOULD-BE Spec $k[x]/(x)$

p272,line -3:that if SHOULD-BE that

p281,10.1.12,line 7: product is $f^{-1}…f^{-1}…$ As $f^{-1}$ SHOULD-BE product is $\pi^{-1}…\pi^{-1}…$ As $\pi^{-1}$

p281,10.1.12,line 8:$…\pi(U_i)$ is a closed embedding SHOULD-BE $…\pi^{-1}(U_i)$ is a closed embedding

p293,10.3.12,(b): on all of $X\times Z$ SHOULD-BE on all of $X\times Y$

p293,line -3:$Y\to X\times Y$ SHOULD-BE $X\times Y\to Y$

p352,line1:induced reduced SHOULD-BE induced

p361,13.1.6:line 4: 13.1.B-13.1.G SHOULD-BE 3.1.B-13.1.H

P368,line-11:with $x\subset$ SHOULD-BE $x\in$

p391,line -1:Supp D (this is not defined explicitly. It may be nice to be inserted at p390,4th paragraph,line 3-4)

p396,14.2.P,line 9:14.2.7 SHOULD-BE 14.2.8

—

June 15, 2013 at 6:28 am

Thanks again! Everything you’ve said has been right. I’ve implemented everything (but not in the version that went public very recently), with the exception of the comment on p. 352, as I defined the phrase “induced reduced subscheme structure” in 8.3.9. Certainly the phrase you suggest, “induced”, is very clear to me, but I’ll stick with “induced reduced” here for the sake of consistency.

June 22, 2013 at 2:42 am

Ok. Thanks.

June 11, 2013 at 4:59 pm

[…] 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 […]