Talk:Algebraic variety

message: It might be nice to have a couple of examples of algebraic varieties in order to make these ideas more concrete and easier to understand. ConfusedGremlin 03:27, 4 June 2007 (UTC)
replies: text: Agreed. Examples before results please. What is the relation to quotient rings ?Rgdboer (talk) 21:52, 10 July 2008 (UTC)

text: I deleted and corrected some of the obvious nonsense.  The circle is explicitly birational to the line by the Weierstrass substitution, so definitely irreducible, but I didn't have time or a reliable source at hand to include this in the article. JackSchmidt (talk) 17:28, 7 May 2009 (UTC)

text: --D.Lazard (talk) 21:06, 14 October 2012 (UTC)
thread_title: Examples

message: The definition as it appears is:
:Let k be an algebraically closed field and let An be an affine n-space over k. The polynomials ƒ in the ring k[x1, ..., xn] can be viewed as k-valued functions on An by evaluating ƒ at the points in An.
This is inaccurate: in order for polynomial functions to be well-defined, we need addition and scalar multiplication of points in the affine space A to be well-defined. This only makes sense if we fix an origin o.
There are two ways to rectify this: 
1. Fix an origin for each polynomial.
1. Fix an origin for A to begin with.
I think that the latter option is the desirable one. However, then there is no need to ask for an affine space. Asking for a vector space over k is enough.
In any case, I don't know enough about algebraic varieties to make this decision. Hopefully someone else does?  — Preceding unsigned comment added by Accidus Benartius (talk • contribs) 10:20, 5 July 2011 (UTC)
replies: text: Actually the affine space  (= set with a free and transitive action of a vector space) structure is irrelevant. One can simply consider the set  kn instead. Liu (talk) 19:26, 12 July 2011 (UTC)

text: Each coordinate of the affine plane (a,b) corresponds to the value of a polynomial in x, y with x fixed to a, y fixed to b. This is desirable because a) functions on the affine space behave like their vector space counterparts, while the actual placement of the coordinates remains free (affine), and b) evaluating every polynomial in K[X] at a point is a homomorphism from K[X]->K, and because points and those homomorphisms are in 1-1 correspondence, the affine plane can then be replaced by that set of homomorphisms. This gap has caused me a lot of trouble. ᛭ LokiClock (talk) 12:54, 2 July 2012 (UTC)
thread_title:  Affine Variety Definition 

message: The following statement is not offered any explanation in the article: "The word "variety" is employed in the sense which is similar to that of manifold". This appears to be grammatically incorrect, but I don't know how to fix it because I don't even know what it's trying to say! Is it merely saying that a variety is a similar object to a manifold in some sense? If so, this is a terrible way to express it. I am sure that any native speaker of English who doesn't already understand what this statement is trying to say will be confused by it. Would anyone care to explain? (For the record, the stuff that follows this statement doesn't clarify it; all it does is explain a difference between varieties and manifolds, which is what led me to the interpretation above)220.245.107.17 (talk) 11:39, 21 September 2012 (UTC)
replies: text: You are right: This is not the meanings which are similar, but the mathematical notions (or concepts). I have modified the formulation accordingly. D.Lazard (talk) 12:55, 21 September 2012 (UTC)
replies: text: This is much clearer, thank you. 220.245.107.17 (talk) 09:34, 23 September 2012 (UTC)
thread_title:  The sense in which the word variety is employed (. . .) 

message: I don't really see why we would have two separate articles here, so I think this other article should be merged in here. Mark M (talk) 15:01, 9 March 2013 (UTC)
replies: text: I agree to merge. How ever, there is a technical difficulty: A good merging implies to give a definition of abstract algebraic varieties in terms of classical algebraic varieties (this is not done in the existing article). IMO, the simplest way (for the broadest audience) is to define an "abstract affine algebraic variety" as the restriction (to be defined) to a Zariski open subset of an affine or projective variety. Then isomorphisms between such objects need to be defined. Then an "abstract variety" is defined as a topological space, equipped with a ring (of polynomial functions) associated to every open set, such that every point has a neighborhood which is isomorphic to an abstract affine variety (construction similar to that of abstract manifolds). IMO, for the broadest audience, references to schemes, should be avoided unless when they are unavoidable or when mentioning generalizations. D.Lazard (talk) 16:01, 9 March 2013 (UTC)
replies: text:  I agree with the merge, but I disagree entirely about avoiding schemes. I think to avoid the actual definition you'd require to use the thing just mystifies the subject, and adds to the fragmentation of algebraic geometry articles across Wikipedia. They're not the plague, they're actually quite nice. In merging, we should motivate the connection between the two, explain how the intrinsic definition subsumes the other. That doesn't mean they have to be defined in terms of classical varieties. That just means they have to be motivated, while the definition itself can and should be given plainly. ᛭ LokiClock (talk) 00:53, 11 March 2013 (UTC)

text: The definition he gave isn't non-scheme-theoretic. It's a description of schemes that doesn't use the word. The Geometry of Schemes attests that schemes can be an introduction to the subject, and not a more abstract way of describing something you'd have to already be familiar with to see in those terms, so it is one foundation of algebraic geometry. Part of what should be immediately represented to the reader in including abstract varieties is that one doesn't need to treat affine varieties and projective varieties as separate kinds of object. What doesn't concern non-mathematicians is up to non-mathematicians to decide. I'm a low-level undergraduate, and I find it hardest to understand things when when the connections to more rigorous definitions are hidden. ᛭ ?LokiClock (talk) 07:37, 11 March 2013 (UTC)

text: That's true. Afterwards, these properties could be related to properties of schemes, which I think is fair. That would solve the another problem, which is that when an object is axiomatized on Wikipedia it often becomes the de facto definition for other articles (e.g., algebraic variety vs. an algebraic set), and from abstract algebraic variety it looks like the definition is highly contingent on the avenue of research. ᛭ LokiClock (talk) 12:07, 11 March 2013 (UTC)
replies: text: I will add two strong reasons to avoid scheme formalism when it is not necessary. Firstly, many important papers on abstract varieties were written before the introduction of the scheme formalism by Grothendieck. Nagata's result is an example. Serre's GAGA theorem in another well known example. Therefore we need a definition that allows to read these papers. Secondly, many people concerned by algebraic geometry are not accustomed to scheme formalism and even do not know it. This is also the case for text books, as one can see by looking at the reference section of algebraic geometry. Among the concerned people not or rarely using the scheme formalism are the specialists in analytic geometry, singularity theory, computational algebraic geometry and real algebraic geometry. This article should also be written for them. D.Lazard (talk) 12:12, 11 March 2013 (UTC)

text: To LockiClock: I agree that scheme formalism is one possible foundation, among several, of algebraic geometry. But, as it is not universally accepted and used, WP:NPOV policy requires that, for the questions for which the other formalisms are yet used, the presentation should take them into account. Fortunately, when things are described in other formalisms, the translation to schemes formalism is usually straightforward for people knowing schemes. The reverse translation is not. D.Lazard (talk) 12:12, 11 March 2013 (UTC)

text: To Lazard and Taku, I did actually think of people working in computational algebraic geometry and real algebraic geometry, but since you've pointed it out I realized everything I've read I translated into schemes implicitly, and in reality there's nothing about those concepts that's dependent on that treatment. ᛭ LokiClock (talk) 19:00, 12 March 2013 (UTC)

text: I was under impression that everyone is for merger. -- Taku (talk) 23:47, 20 March 2013 (UTC)
replies: text:  We were, but I was reviewing the merger help pages, and it mentioned merging pages that were unlikely to develop into their own articles. My question now is whether, when we need to develop the subject of an abstract algebraic variety, we are actually needing to develop this article. I'm getting that impression from how we're talking about it, but I want to make sure we're on the same page. ᛭ LokiClock (talk) 00:59, 21 March 2013 (UTC)
thread_title:  merge in [[Abstract algebraic variety]] 

message: Thinking again about the preceding discussion, it appears to me that a single article covering all the material from affine varieties to abstract varieties and scheme in not a good idea. In fact, the difficulty of algebraic geometry is that it has been formalized with different languages: the classical language of affine and projective varieties and several languages for abstract varieties: roughly speaking, Weil's formalization, Serre's view as ringed spaces over the usual topology, well suited for GAGA and schemes. The most convenient language depends on the questions that are considered, and several of these languages are yet commonly used. Even Weil's language is yet needed: I do not know a simpler way to define a "generic change of coordinates" of the affine space.
On the other hand, the part of the article devoted to affine and projective varieties is a stub, which needs to be widely expanded. For example, nothing is said about the dimension or the degree of a variety nor on the fact that the Zariski closure of the image of an irreducible variety by a morphism or a rational map is an irreducible algebraic variety. IMHO, all these issues may not be solved in a single understandable article, and I suggest two interlinked articles:
*Algebraic variety (affine and projective): (I put "affine and projective" after "algebraic variety" for an easier search) dedicated to the classical language. This article could be mainly based on the relevant material in  (this is the basic reference for applied algebraic geometry)
*Algebraic variety (abstract), mainly based on Hartshorne's book, introducing and using a more abstract language.
D.Lazard (talk) 21:30, 21 March 2013 (UTC)
replies: text: You are aware of the articles Affine variety, Projective variety, and Quasiprojective variety? Are you proposing to add yet another article? I don't think that's a good idea; I think the algebraic variety article is a perfect place to discuss how these concepts are related to each other; and we could even have a section about how it relates to the scheme-theoretic definition. Splitting into two will only confuse the situation, since the concepts are so closely related. Mark M (talk) 09:44, 22 March 2013 (UTC)
thread_title:  Merge or split? 

message: Abstract variety is a duplicate of the recently merged abstract algebraic variety. ᛭ LokiClock (talk) 00:09, 22 March 2013 (UTC)
replies: text: Agree; I've redirected it here. Mark M (talk) 09:54, 22 March 2013 (UTC)
thread_title:  Duplicate article: Abstract variety 

message: This is a surprisingly hard question to answer. I know it is a particular kind of a "scheme". But that definition would not be neutral enough to Wikipedia, it seems. (Also, such a definition is technically problematic when we need to define, say, a complex-analytic variety by replacing polynomials by holomorphic maps.) I believe the official definition of an algebraic variety should be a particular kind of a locally ringed space (i.e., it is locally an affine variety, irreducible, separated, finite-type), no "scheme" is needed in this definition. But I don't have a source or maybe it's Serre's definition? (The lack of the definition of an algebraic variety creates a host of issues, like our inability to give a clear definition of a regular map.) For example, I think we should at least define an algebraic prevariety; an algebraic variety would then be defined as a separated algebraic prevariety. -- Taku (talk)
:IMO, this question is a non sense: Over the 20th century, the concept of variety has evolved by successive generalizations that include the preceding concepts as particular cases. These generalizations were required for solving important conjectures. For example, Grothendieck explains explicitly in the introduction of EGA that one of the motivation was the future proof of Weil's conjecture. Each generalization contains the previous notions as particular cases. However these generalizations are very abstract and difficult to understand for people that have not yet considered the problems for which they have been introduced (some say that they are "esoteric"). Moreover they are unadapted for many classical questions. Thus giving a unique very abstract definition (as a scheme or as a locally ringed space) would make impossible to use algebraic geometry for many possible users (such as in computer geometry or cryptography). For example, given a projective variety by its equations, how do you use scheme theory for computing its dimension? (the best known method is to compute a Gröbner basis and deduce the Hilbert series, none requiring to know what is a scheme). Also, ringed spaces and schemes are almost never used in real algebraic geometry. IMO, the article is correct as it is, by giving successive generalizations, more or less in their historical order. Trying to provide a unique definition that works in every situation would clearly be WP:OR. D.Lazard (talk) 12:39, 4 February 2015 (UTC)
thread_title:  What is an algebraic variety? 

message: I suggest that mergeing the algebraic manifold and the singular point of an algebraic variety. Because I think it makes sense to explain the algebraic manifold while explaining singular points and smooths. SilverMatsu (talk) 04:59, 17 February 2023 (UTC)
replies: text: I am not sure if that's good idea, because while algebraic manifolds are studied in differential geometry or real algebraic geometry, singularities of varieties are studied over arbitrary fields, i.e., the contexts differ. I do wonder, however, if we need "algebraic manifold" as a separate article from this one. -- Taku (talk) 08:10, 17 February 2023 (UTC)
replies: text: Thank you for your reply. I agree with that; . It seems that Nash's result is explained in Nash manifold. Just to let you know, the analytic manifold seems to have been merged into the Differentiable manifold. --SilverMatsu (talk) 01:57, 18 February 2023 (UTC)
thread_title:  Merger proposal 

message: The redirect Differential algebraic variety has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached.  1234qwer1234qwer4 22:44, 26 March 2024 (UTC)
thread_title:  "[[:Differential algebraic variety]]" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]