Talk:Algebraic geometry

Edit waring

User Wcherowi has reverted another sourced edit from me on the Sharaf al-Dīn al-Ṭūsī page without providing any source but his own opinion... Please user Wcherowi, just have a look at the Roshdi Rashed's article on Wikipedia and tell me where you can see that that guy is "biased" ?

Bloated

The sections of the article have become too long to the point of being offputting to a new reader, and should be reorganised. Similarly, the first section should be an introduction to the most important concepts/results in algebraic geometry and relations to other fields, not (as it is now) the first quarter of an intro AG course reproduced verbatim. - 30 July 2020 — Preceding unsigned comment added by 192.76.8.75 (talk) 13:03, 30 July 2020 (UTC)

Developpement of Algebraic geometry

In the section "history" of this article, it's stated that "Each of these early developments in algebraic geometry dealt with questions of finding and describing the intersections of algebraic curves.", but this is not sourced... Here is a source which writes that algebraic geometry was ingurated with Sharaf al- din al-Tusi:

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_mathematics.html

"Sharaf al-Din al-Tusi (born 1135), although almost exactly the same age as al-Samawal, does not follow the general development that came through al-Karaji's school of algebra but rather follows Khayyam's application of algebra to geometry. He wrote a treatise on cubic equations, which represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry."

The source cites Roshdi Rashed who is a prominent academic (bronze medal of French CNRS...) for this topic. So i'm about to revert the ambiguous sentence about the "early developpement in algebraic geometry" and replace it by the source above but i would like to have other's editors opinion please. 2A01:E34:EE9D:A200:90E9:FDE6:16CF:3B09 (talk) 16:23, 10 September 2017 (UTC)

This statement has been in the article and was removed in part because it was a copyright violation. Also, Rashed's opinion in this matter is clearly biased as he did the modern translation of al-Tusi's works. You would need to find a source, preferably by an expert in algebraic geometry, that supports this position. --Bill Cherowitzo (talk) 18:05, 10 September 2017 (UTC)

I don't understand why briefly citing a source is a copyright violation... I don't share your POV about Rashed, it's true that some others scholars (Berggren, Hoogendijk) do not share his position about some medieval muslim contributions, but who can say which one is right ? when i look at Rashed's curriculum vitae, it's pretty impressive and i'm sorry, but i can not believe your words about him. The question is "is Rashed a reliable source for this topic?" and the answer is, without a doubt, YES. And even if you're right, and Rashed is biased about this subject, have a look at this :

http://en.wiki.x.io/wiki/Wikipedia:Neutrality_of_sources

It's said that "Reliable sources may be non-neutral"...

The fact is that this article contains some contradictions and we should try to solve them instead of exposing our own POV... 2A01:E34:EE9D:A200:C5BC:E8D1:72E7:B92F (talk) 22:29, 10 September 2017 (UTC)

Tropical geometry should be mentioned

The subject of tropical geometry has now developed to such an extent that it deserves some mention in this article, and this article should link to the one on tropical geometry. The phrase actually occurs in the present version, but it appears without explanation. What is needed are more elementary remarks, perhaps also a simple example and picture. Ishboyfay (talk) 06:09, 17 April 2020 (UTC)

More generally, the subfield of combinatorial algebraic geometry could be mentioned. Ishboyfay (talk) 06:11, 17 April 2020 (UTC)

I do agree that the field has developed large enough to warrant some section. But it should not be too long because of WP:DUE. I suppose one picture (but no more!) can be allowed, in my humble opinion. —- Taku (talk) 06:29, 17 April 2020 (UTC)

Logarithmic algebraic geometry is another subfield that deserves its own section. -- Taku (talk) 02:37, 22 April 2020 (UTC)

Suggestion for a large reorganisation

As noted elsewhere, the current article is a bloated mess. What do people think about the following alternative organisation?

1) Cut the introduction to 1/2 length, change it to mostly contain examples and pictures. The bullet point list of connections to other subjects is good, keep it, but just illustrate each with a picture/example. For instance, Mumford type pictures of Spec (...), classical conic pencils, Riemann surfaces, etc.

2) Replace the "basic notions" sections with a section giving the classical definition of variety, but also of a scheme, a stack, and prestack. Make the latter definitions ultra-concise, with the main emphasis being examples.

3) Before or after this section, have a section devoted to "the most important" theorems in AG, (Grothendieck)-Riemann-Roch, Chow's theorem, ... [this list is really long and should be completed by many people, but each theorem should be concise so as not to induce bloat]

4) Have a section explicitly talking about connections to other subjects, e.g. Atiyah-Bott's Yang Mills paper, ...

5) The current history section is good, just needs to have a few more mathematical details, pictures, links, etc. MeowMathematics (talk) 23:34, 29 September 2023 (UTC)

I am very dubious with this project:

Item 1: for cutting the introduction to 1/2 length, you must state what should be removed. Presently there are 5 paragraph: a short definition, a description of the object of study, a description of the context, a list of subareas and a paragraph that may be viewed as an explanation of the importance of scheme theory. Personally, I would have written this paragraph in a completely different way, but it remains that scheme theory must be mentioned in the lead. So, I do not see what could be cut here. Also, the introduction of the article must be a summary of the content; so examples and illustrations of subareas belong to the corresponding section.

Item 2: I agree that Section § Basic notions must be changed, but not by increasing its technicality as you suggest. Instead most of its content must be left to linked articles, and the section must be focused on the minimum that is needed for understanding the remainder of the article. For example, morphisms, regular functions, birational equivalence do not belong to this article, but the correspondences points/maximal ideals and subvarieties/prime ideals (Hilbert's Nullstellensatz) must be mentioned, as well as the Zariski topology. Hilbert's Nullstellensatz is fundamental as it is the starting point of the emergence of algebraic geometry as an autonomous field, and the Zariski topology is the most elementary concept that allowed scheme theory.

Item 3: There are too many important theorems in algebraic geometry to be listed in this article. Even for an autonomous list article, I doubt that you can establish a selection criterion that makes such an article acceptable.

Item 4: I understand that as a requirement to expand section § Applications and to split it in several subsections, including applications to theoretical physics. If I am right, I agree.

Item 5: Section § History is not really good: it lacks the information that algebraic geometry was not an autonomous field before the 20th century, and that Hilbert's Nullstellensatz is the fundamental result that allowed the emergence of algebraic geometry as an autonomous field. D.Lazard (talk) 11:11, 30 September 2023 (UTC)

I must slightly moderate my assertion on the Nullstellensatz as the starting point of algebraic geometry: there are results of algebraic geometry that date from the second half of the 19th century, such as Bézout's theorem and the results of Italian school of algebraic geometry, but, it is the Nullstellensatz that alloved eventually dealing with non-genericity and singular varieties Indeed, Bézout stated his theorem for "general" equations only ("generic" in modern terminology), and Italian geometers studied mainly smooth algebraic surfaces. It is also Nullstellensatz that allowed replacing eventually methods of complex analysis by purely algebraic methods. In summary, my (unsourced) opinion is that Nullstellensatz is the starting point of the separation between complex geometry and algebraic geometry. This is what I meant with the word "autonomous". D.Lazard (talk) 10:58, 1 October 2023 (UTC)

Thanks for the reply.

1) I think a lot of the introduction would work better moved to later sections. The main issue imo is that currently it's very long (~600 words), mostly consists of namedropping objects/subjects (beginners, the people for whom the introduction is presumably written, will not recognise most of these), and no examples or pictures. I would propose keeping the 1st paragraph, then for the rest carefully picking out the most important/representative examples, and explaining them in examples and/or pictures in a way that e.g. someone just starting a first AG course could get the jist of (or if we choose to include more advanced topics, someone who's just on the edge of being able to learn those). The rest is good, but should be moved to later sections.

2)i) It's good we agree that those sections should be cut.

2)ii) If you're going to the trouble to mention Nullstellensatz, I don't think we're spending much more of our complexity budget by introducing a) the reconstruction of a topological space from its ideals, as a motivator together with the Nullstellensatz to b) the definition of Spec A.

2)iii) Stacks (moduli stacks, examples from geometric rep theory) are such a fundamental part of geometry now that I think at very least they a section later on, though I agree the first post-intro section is probably too soon.

3) OK I agree, how about something more like "most important connections with other subjects" (which is a bit more objective, just take a look at subjects by citation volume), and for these there are often foundational theorems that could or could not be worth discussing. By the way, in the areas of AG that it includes I would suspect the current article does not fit the selection criterion - it's mostly classical and computation AG at the moment, which is extremely far from what the current or historical activity in the subject is/has been. I'm not saying this to criticise the article, rather that if we try to move it more in that direction we're likely to improve it wrt the selection criterion as applied to what areas we cover. n.b. this is where I think we should put a lot of the stuff currently in the introduction.

4) That would be great, I agree!

5) Ah OK, I don't really have much of an opinion on this. Ideally a historian of mathematics could have a look at it. MeowMathematics (talk) 17:36, 2 October 2023 (UTC)