Talk:Algebraic variety

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Examples

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)[reply]

Agreed. Examples before results please. What is the relation to quotient rings ?Rgdboer (talk) 21:52, 10 July 2008 (UTC)[reply]

N.B. We need that second example corrected, it's plain wrong. The circle lying in the complex numbers IS irreducible. The two sets that the circle is the union of aren't algebraic, they can't be. A subset X of the complex numbers is irreducible with respect to the real numbers if and only if R[X] is an integral domain - R is the real numbers. One may write any element of R[X] as "a + bx + cy + f(y)" where a,b,c constants in R, x,y are the coordinate functions in R[X] and f is a polynomial in one variable, here in "y". It's plain to see, even with x² = 1 - y², that one has no zero divisors and so R[X] is certainly an integral domain. The circle is an affine variety. —Preceding unsigned comment added by 92.25.162.33 (talk) 16:24, 7 May 2009 (UTC)[reply]

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)[reply]

There should be an example of an algebraic variety that cannot be reduced to one polynomial. For example, in R³, the intersection of x² + y² + z² = 4 and (x - 1)² + y² = 1 is equivalent to the intersection of the second polynomial with z² + 2x = 4, but it cannot be reduced to one polynomial. --Moly 15:58, 14 October 2012 (UTC) — Preceding unsigned comment added by Moly (talk • contribs)

Done--D.Lazard (talk) 21:06, 14 October 2012 (UTC)[reply]

Affine Variety Definition

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, ..., x_n] 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)[reply]

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)[reply]

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)[reply]

Would it be possible to construct the equation in such a way that we could add a link to the definition of set theory symbols, not throughout the article, but just the first instance, such as ∈ and ∨ in the equation under "Affine Variety Definition"? Just being pedantic, I guess, I knew decades ago what the symbols mean, but, after all, it is a definition, particularly the AND which some might see as a 'V'. Hpfeil (talk) 16:07, 1 September 2013 (UTC)[reply]

merge in Abstract algebraic variety

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)[reply]

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)[reply]

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)[reply]

A larger question to which I don't know the answer is this: is the theory of schemes now the foundation of algebraic geometry in the same sense, say, the (axiomatic) set theory is the foundation of mathematics? If yes, it would simply create more confusion if we avoided schemes. It is also important to remember (and I keep forgetting) that we're not writing solely for math graduate students and beyond. (But maybe this "abstract" stuff may only concern pure mathematicians.) By the way, it is important to realize that it does matter if one is doing things in the terms of varieties or schemes. For example, the intersection in the sense of varities would not be the same as that in the scheme sense. I suppose I'm mainly with D. Lazard: we should at least give the non-scheme theoretic definition of abstract varieties. (Personally, I'm very interested in historical development of the notion; e.g., the works of Nagata. But I'm too busy with other stuff in Wikipedia.) -- Taku (talk) 01:24, 11 March 2013 (UTC)[reply]

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)[reply]

While we're at it, I think we should also merge in Quasiprojective variety. Harris, for example, uses the word "variety" to mean a "quasi-projective variety". I agree that the definition involving schemes shouldn't be avoided entirely, but at the same time, I think we should introduce the concept without using the word scheme. For example, Hartshorne, p.105, defines a variety to be an "integral separated scheme of finite type over an algebraically closed field" (not a good introductory definition!).. but earlier, on p.58, he defines the term "abstract variety" without using the word scheme (similar to D.Lazard's suggestion above). Without using the word "scheme", the question of whether of whether you can glue together quasi-projective varieties to get something that's non-quasi-projective is natural, and non-obvious. As stated in Hartshorne p.105, Nagata was the first to find an example of a non-quasi-projective variety. This is the kind of fact that is perfectly suited for an article called "Algebraic variety". Mark M (talk) 10:37, 11 March 2013 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

Replies to several points:

I'm not too sure about the merger of quasi-projective varieties. To some workers in algebraic geometry, "quasi-projective" is an important question, I imagine. The discussion on a "quasi-projective scheme" also fits naturally into the article.
To LokiClock, I agree a scheme is a conceptually much simpler: an affine scheme is Spec of a ring and a projective scheme of Proj of a graded ring; couldn't be simpler. If one is learning the subject for the first time, then the "Geometry of schemes" style may be simpler than Hartshorne since a variety, abstract one at that, get defined only once. The complication here is that we're not writing a textbook and the article, at least part of it, has to be suitable for readers of varying backgrounds. Schemes are clearly not suitable to some readers.
To Lazard, it is news to me that the translation is straightforward :) I have had so much trouble going from one formalism to another: evil is in the details if the intuition is not a problem. Therefore, a large portion of this article "should" discuss how one goes from Weil's universal field stuff to varieties that are actually schemes, etc.

-- Taku (talk) 18:11, 12 March 2013 (UTC)[reply]

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)[reply]

I was about the merge, but obviously Abstract algebraic variety can develop as a full article. The problem is, right now it's exactly the content that needs to be merged into here. It seems like the least messy policy in the long term is to duplicate the content there as a section here. ᛭ LokiClock (talk) 21:38, 20 March 2013 (UTC)[reply]

I was under impression that everyone is for merger. -- Taku (talk) 23:47, 20 March 2013 (UTC)[reply]

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)[reply]

That depends on the length of this article, I think. When this article gets too long, the natural step is to split off some materials to a separate article. Will it happen? Don't know, and my attitude is not to worry about this sort of matter since developing materials (e.g., sourcing, writing, quality assurance, etc) is "much" harder than moving stuff around. -- Taku (talk) 13:22, 21 March 2013 (UTC)[reply]

Irreducible?

Perhaps it is better to drop the convention that algebraic varieties are irreducible in this article. Firstly, there's not really a consensus in the literature (for example Harris, or Springer's Encyclopedia of Math entry, written by Dolgachev). And secondly, there's no natural name for reducible algebraic varieties. Because (as far as I can tell) the term "algebraic set" isn't widely used (maybe "affine" or "projective algebraic sets" are), and it wasn't defined in the article anyway. The definition (which I removed) was that an algebraic set was the set of solutions to polynomial equations. But what does that mean? Are non-quasi-projective varieties examples of algebraic sets or not? On the other hand, the phrase "irreducible algebraic variety" could be used whenever we want to emphasis that some variety is irreducible. Note that the MOS says the convention is to assume algebraic varieties are irreducible (which seems reasonable as a default convention, outside of this article). Thoughts? Mark M (talk) 16:48, 21 March 2013 (UTC)[reply]

It seems that you think the merge as making affine and projective varieties special cases of abstract or quasi-projective varieties. This can not be done: Most readers are not specialists of algebraic geometry, and the article should be written for them. Thus, the article should begin with affine varieties, then projective, then quasi-projective and abstract varieties. I agree that "algebraic set" is mainly used in the affine and projective case. But most users of algebraic geometry, like specialist of mechanics or physicists, do not know anything else than affine and projective cases (and few of them know of the projective case). This makes the above grading of technicality essential. Moreover, in most applications, algebraic varieties are given by equations, and deciding if a given set of equations defines an irreducible variety or not is a hard task. This makes essential to warn the reader that not every set of equations defines an irreducible variety. Using a few lines in the lead to clarify the terminology seems the best way for such a warning. Moreover, "algebraic set" must appear in the lead in boldface, because algebraic set redirects here, and it would be a non sense to write a separate article. D.Lazard (talk) 18:49, 21 March 2013 (UTC)[reply]

I disagree that we must add a definition of algebraic set to the lead. Perhaps simply saying that any set of solutions in A^n of some collection of polynomial equations is called an algebraic subset of A^n. The term "algebraic set" on its own, for example, is used in two different ways in Hartshorne, so we should be careful about how we define it.

I don't understand your other suggestion.. affine and projective varieties are special cases of abstract and quasi-projective varieties. Why can't we say so? I think we should place emphasis on the importance of affine and projective varieties.. and I agree we should introduce those first, and then introduce quasi-projective, and then abstract varieties. So, I'm not sure that we are actually disagreeing on anything here..? Mark M (talk) 09:37, 22 March 2013 (UTC)[reply]

Merge or split?

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 David A. Cox (1997). Ideals, Varieties, and Algorithms (second ed.). Springer-Verlag. ISBN 0-387-94680-2. Zbl 0861.13012. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) (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)[reply]

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)[reply]

Duplicate article: Abstract variety

Abstract variety is a duplicate of the recently merged abstract algebraic variety. ᛭ LokiClock (talk) 00:09, 22 March 2013 (UTC)[reply]

Agree; I've redirected it here. Mark M (talk) 09:54, 22 March 2013 (UTC)[reply]

What is an algebraic variety?

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)[reply]

Merger proposal

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)[reply]

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)[reply]

Thank you for your reply. I agree with that; I do wonder, however, if we need "algebraic manifold" as a separate article from this one.. 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)[reply]