Talk:Resolution of singularities

The article begins thus:

In algebraic geometry, the problem of resolution of singularities asks whether any algebraic variety has a non-singular model (a non-singular variety birational to it).

"Any" is often ambiguous. Does it ask

• whether there is any algebraic variety that has a non-singular model; or
• whether it is the case that any (i.e. every) algebraic variety has a non-singular model?

I don't know algebraic geometry, but I know it's not unusual for mathematicians to use "any" while overlooking this potential misunderstanding. In about 2002, the article on the compactness theorem in logic said that that theorem states that a set of sentences "has a model if any finite subset has a model". That's begging to be misunderstood! If the second meaning above was intended, then just changing "any" to "every" would fix the problem. Michael Hardy 03:48, 29 March 2007 (UTC)

I just noticed that nobody had fixed this, 2.5 years later. Of course, the second meaning is what is meant. Now changed. Artie P.S. (talk) 08:25, 29 October 2009 (UTC)

"Any" always means "any random". If it were about existence, one would use the term "some". The problem with "every" is for example: "Every algebraic variety X has a non-singular model Y." One could think that Y is independent from X. Usually, people use "any" and "some". Spaetzle (talk) 09:17, 14 July 2015 (UTC)

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This page uses the term "blowup" without first defining it. I've created a link, directed to "blowing up". Should a definition be added in this article? Michael Hardy 00:26, 18 April 2007 (UTC)

Fact tag

An anonymous editor has added a fact tag to the last sentence of the lede (which says something along the lines of "resolution of singularities in characteristic p is an important open problem in dimension at least 4". As I understand it, in uncontroversial articles of this kind the convention is that it's ok to write things like this (for which, although generally accepted as true by experts, it may be difficult or impossible to find an explicit citation) without violating WP:NOR. More precisely, it seems as though the exposition of many mathematics articles would be worsened if all such statements had to be removed.

Of course this must be a much wider issue across maths articles, so I'm sure people have opinions about how best to proceed. If you have any opinion about this, please let me know. Artie P.S. (talk) 08:21, 29 October 2009 (UTC)

As often happens, this is a writing issue, really. The word "important" is just lazy here. It is an open problem; research is done on it, and that means it is significant. Rephrase as something like "resolution in char p is an open problem in the general case; the currently-available techniques only work for threefolds". This is uncontentious material suitable to stand in the lede. Charles Matthews (talk) 09:54, 29 October 2009 (UTC)

(ec)Surely there are other citations for this one, and no harm is done by making common knowledge and commonsensical statements like it, but here's something Grothendieck said, indirectly, about its importance - and who would dispute? "Alongside the problem of resolution of singularities, the proof of the standard conjectures seems to me to be the most urgent task in algebraic geometry."[1] (last sentence of paper, p.198).John Z (talk) 10:04, 29 October 2009 (UTC)

About the Proof in Characteristic Zero

(This is just a plan to add a section to the article with a sketch of the proof. It will probably take several days so I'm placing it here first. Feel free to add and make some changes.)

There are many constructions of strong desingularization but all of them give essentially the same result. In every case the global object (the variety to be desingularized) is replaced by local data (the ideal sheaf of the variety and those of the exceptional divisors and some orders that represents how much should be resolved the ideal in that step). With this local data the centers of blowing-up are defined. The centers will be defined locally and therefore it is a problem to guarantee that they will match up into a global center. This can be done by defining what blowings-up are allowed to resolve each ideal. Done this appropriately will make the centers match automatically. Another way is to define a local invariant depending on the variety and the history of the resolution (the previous local centers) so that the centers consist of the maximum locus of the invariant. The definition of this is made such that making this choice is meaningful, giving smooth centers transversal to the exceptional divisors.

In either case the problem is reduced to resolve singularities of the tuple for by the ideal sheaf and the extra data (the exceptional divisors and the order, $d$, to which the resolution should go for that ideal). This tuple is called a marked ideal and the set of points in which the order of the ideal is larger than $d$ is called its co-support. The proof that there is a resolution for the marked ideals is done by induction on dimension. The induction breaks in two steps

1. Functorial desingularization of marked ideal of dimension $n-1$ implies functorial desingularization of marked ideals of maximal order of dimension $n$.
2. Functorial desingularization of marked ideals of maximal order of dimension $n$ implies functorial desingularization of (a general) marked ideal of dimension $n$.

Where we say that a marked ideal is of maximal order if maximal order if at some point of its co-support the order of the ideal is equal to $d$. —Preceding unsigned comment added by Franklin.vp (talk • contribs) 01:48, 3 November 2009 (UTC)

complete variety

Why it is asked for the resolution to be complete? Isn't the affine space its own resolution? Abisharan (talk) 15:21, 22 June 2010 (UTC)

What is asked usually is that the morphism to be proper to avoid just taking the inclusion of the set of regular points as a resolution. Abisharan (talk) 15:23, 22 June 2010 (UTC)

Blowing up the most singular points does not work

In this section it is given Bierstone-Milman as an algorithm blowing-up the worst singularities. This is kind of misleading in two ways. One is that all the other (general) algorithms essentially do the same since even though an invariant is some times not used to indicate the center of blowing-up as its maximum locus it always is the case that such and invariant can be given. The other reason is that thinking of it as blowing the worst singularities in some sense hides the fact that the algorithms use information not only about the variety (the strict transform in each stage [or year]) but also about the blowings-up previously performed. Thoraeton (talk) 18:16, 11 July 2010 (UTC)

Weak and Strong resolutions.

That terminology is not quite settled. It is true that it is used in the literature but is not uniform. Well, the strong adjective maybe, but it is usually applied to distinguish a resolution that is functorial from those that are not. But even in that case there are many ways to make the result stronger. Since Wikipedia doesn't have to say everything, I guess it is better not to introduce confusion by stating definitions that are not really solid in the grown or that change from source to source. Thoraeton (talk) 19:32, 12 July 2010 (UTC)

Graph labeled incorrectly.

The graph at the top right of the article is the graph of X:={x^3-y^2=0}. The owner of the graph should correct this. KYSide (talk) 17:56, 20 October 2011 (UTC)KYSide

English language

In the section "Method of proof in characteristic zero" there are several meanings which are not valid English:

"In every case the global object (the variety to be desingularized) is replaced by local data (the ideal sheaf of the variety and those of the exceptional divisors and some orders that represents how much should be resolved the ideal in that step). With this local data the centers of blowing-up are defined. The centers will be defined locally and therefore it is a problem to guarantee that they will match up into a global center. This can be done by defining what blowings-up are allowed to resolve each ideal. Done this appropriately will make the centers match automatically."

I guess, "Done this appropriately" should be "Done appropriately, this" and I will fix that one. But I don't understand what the author wants to tell us with "how much should be resolved the ideal in that step". Can somebody please fix that sentence? Spaetzle (talk) 09:24, 14 July 2015 (UTC)

Picture in the top right corner

In the picture "Strong desingularization of X:=(x^2-y^3=0) \subset W:= \mathbf{R}^2" it seems that the labels x and y in the top coordinate system should be exchanged. — Preceding unsigned comment added by 131.220.184.95 (talk) 09:59, 1 October 2015 (UTC)

Swapped the labels of the axes in the image.

If someone wants to add the image to the article, here it is.

Fixed image here — Preceding unsigned comment added by 76.10.163.92 (talk) 17:25, 23 March 2016 (UTC)

Newton citation incorrect

At present, the Newton method citation is for the wrong letter, [[2]]. Flagging this in the hope that it is resolved.

Main image is incorrect

The last part of the main figure showing the resolution of the cuspidal curve is incorrect: it is missing an exceptional curve. There should be a total of 4 curves in the final image, see Hartshorne's "Algebraic Geometry" page 392, example 3.9.1, figure 20. Dzackgarza (talk) 02:07, 1 December 2022 (UTC)