Talk:Smooth completion

Source of material

Most of the material here is from Talk:Algebraic_geometry#Improving_sections and is due to users Ozob and Liu. Tkuvho (talk) 13:40, 6 February 2012 (UTC)

Terminology

Either I do not understand the concept or I disagree with the terminology: "completion" seems to imply that the curve is embedded in its completion. This is not always the case, because of the use of blows up: There is a surjection, not always injective from the "completion" onto the curve. How is it possible to embed the crossing point of $V(x^{3}-x^{2}+y^{2})$ in a non singular curve?

I was suggesting to rename this page Desingularization of curves, but looking on the page Desingularization I have found that the section on curves is rather complete, although it does not take the problems of rationality into account.

By the way, it seems that the problem with imperfect fields is that, if the field k is not perfect, a limit of points having their coordinates in k may not have its coordinates in k. Thus the "smooth completion" of a curve defined on k may be not defined on k, and one may have to extend k to get it. Is there something more to add (at WP level) for imperfect fields?

D.Lazard (talk) 19:05, 5 January 2012 (UTC)

As far as the terminology is concerned, I agree that the curve is not contained in its desingularisation. However, I have the impression (also from the discussion at talk:algebraic geometry) that it is accepted terminology. Certainly if the editors come to an agreement that the term is wrong, the page should be redirected. As far as imperfect fields, I hope Ozob or Liu will comment. Tkuvho (talk) 19:40, 5 January 2012 (UTC)

Perhaps the term "smooth completion" is only applied when the original affine curve was already regular at each point (note that the closure in projective space may be singular, as typically happens, for example, for hyperelliptic curves). In that case I think it should be contained in the "smooth completion". Tkuvho (talk) 19:42, 5 January 2012 (UTC)

The page Resolution of singularities is rather technical and gives very little in the way of details on curves. I think there is room for a more leasurely discussion at a more elementary level. Tkuvho (talk) 21:04, 5 January 2012 (UTC)

• Merge This page is still too short to be an article but based on the arguments given at Wikipedia:Articles for deletion/Smooth completion i suggest a merge rather than an outright delete — Preceding unsigned comment added by Wikishagnik (talk • contribs) 07:41, 5 February 2012 (UTC)

Comment on completion

The word completion is standard in the sense that every separated algebraic variety (over an arbitrary field) is an open dense subset of a proper (or complete in the old terminology) algebraic variety. This is a theorem of Nagata. If one starts with a smooth algebraic variety, one can wonder whether a smooth completion exists. This is true if the base field has characteristic 0, this is Hironaka's theorem on the resolution of singularities (take any completion and solve the singularities contained in the boundary). In higher dimension (at least 3), there is no canonical smooth competion. In positive characteristics, it is conjectured but still unproven that resolution of singularities exists.

The situation for curves is a little better. If we start with a (separated) smooth curve C, the normalization process gives a regular completion which is unique and the inclusion of C in any proper curve C' extends to a morphism from the regular completion to C'. Note that the regularity is a weaker property than smoothness. If the base field is perfect, then regularity is equivalent to smoothness and we have a well defined smooth completion. If the base field is imperfect, it can happen that the regular completion is not smooth, hence no smooth completion exists in this case. Note also that a proper curve is always projective (this is no longer true in dimension bigger than 1). The notion of smooth completion is interesting because it allows to consider a smooth curve as a smooth projective curve minus a finitely many points.

For non-smooth curves, it is probably unreasonable to use the name smooth completion. Liu (talk) 22:55, 6 February 2012 (UTC)

Thanks, that's fabulous. Could you perhaps incorporate some of this information into the article? Tkuvho (talk) 08:14, 7 February 2012 (UTC)

I disagree with the term "smooth completion" for a possibly non-smooth curve. I don't have access to Griffiths's reference here. What is said in the introduction is rather a "smooth model" of the function field of an integral curve. Liu (talk) 09:59, 7 February 2012 (UTC)

The question is not so much whether one finds the terminology logical, but whether it is widespread or not. If it is used routinely in Hartshorne then certainly a page here is justified. We can mention the fact that in the singular case the affine curve will not be included. Do we stop using the ${\frac {dy}{dx}}$ notation just because it is illogical? Tkuvho (talk) 13:14, 7 February 2012 (UTC)

I just checked with Griffiths's reference, he is talking about smooth completion of smooth quasi-projective varieties. Liu (talk) 21:40, 7 February 2012 (UTC)

Projective closure vs. smooth completion

From the previous posts, I understand that the lead of this article should begin with something like

The projective closure of an affine smooth algebraic variety is generally not a smooth variety. It is thus a relevant problem to know if every smooth variety may be embedded as a dense subset in a complete algebraic variety...

The point is that the page Projective closure does not exists, although this notion is, in my opinion, more important then "smooth completion". There are two ways to solve this issue. Either to create the lacking page. Or to merge the two notion in a single page Projective closure and smooth completion with redirects for each notion. My opinion is that the second solution is better for the users, as it is difficult to understand the question of "smooth completion" without knowing of "projective closure". D.Lazard (talk) 13:34, 7 February 2012 (UTC)

An easy and intuitive point of entry to the subject is provided by smooth affine curves, which have a smooth completion and moreover the latter is unique. If one's goal is to write an accessible article that gives a non-expert reader a point of entry into what is a very technical subject, certainly "smooth completion" is it. Tkuvho (talk) 13:46, 7 February 2012 (UTC)

separated

Is "separated" the same as separable? Tkuvho (talk) 12:17, 13 February 2012 (UTC)

That does not seem to make sense. Could one provide a link to the appropriate page? Tkuvho (talk) 12:18, 13 February 2012 (UTC)

Done (even though the linked page is not very satisfactory). An algebraic variety X is separated if the diagonal in X x X is a closed subset. Affine varieties as well as projective varieties are separated. Subvarieties of separated varieties are separated. Complete varieties are separated by definition. So a variety embeddable in a complete variety is necessarily separated. Nagata's theorem says this condition is sufficient. Liu (talk) 19:38, 13 February 2012 (UTC)

Is it correct to say that a variety is separated if and only if it is "complete" in the metric sense with respect to an appropriate metric? Tkuvho (talk) 13:34, 14 February 2012 (UTC)

The Zariski topology is essentially never metric (not T_1). The definition of separatedness can be understood as follows. An usual topological space X is separated (i.e. Hausdorff) iff the diagonal map X -> XxX (where the righthand side is endowed with the product topology) has closed image. An algebraic variety X is called separated if the diagonal map X -> XxX (where the righthand is endowed with the Zariski topology) has closed image. On XxX the Zariski topology is finer (has more closed subsets) than the product topology. In fact the image of the diagonal map is almost never closed for the product topology. This explains why X can be separated as algebraic variety, but no separated in the usual topological sense. Liu (talk) 22:06, 14 February 2012 (UTC)