Talk:Algebraic geometry/Archive 1

Format of references
I've changed the references to use the book citation format

kris 02:45, 10 April 2006 (UTC)

I have removed this:

'Closely related are commutative C* algebras. For example, the example above corresponds to the commutative unital C* algebra generated by x,y and z subject to the relation $$x^2+y^2+z^2-1=0$$.'

There is no particular advantage to using the Gelfand representation to the spectrum of a ring for making this point - unless it happens to be more familiar to a given reader.

Perhaps some comment about this belongs in section 5.

Charles Matthews 11:50, 10 Sep 2003 (UTC)


 * This article seems to me a little breathless -- especially the introduction. Rick Norwood 22:56, 8 January 2006 (UTC)


 * That seems wholely appropriate - I'm taking a course in this subject right now, and it seems pretty breathless, especially the introduction :P Indigenius 13:17, 12 October 2006 (UTC)

Great Introduction
I enjoyed the introduction given; however, the claim made there that

"When there is more than one variable, geometric considerations enter and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution"

was not, to my view, substantiated in the text. Would you give an example where knowing the geometry of the zero locus is important?

For example: (1) does curvature plays a role? (2) Is there a concrete application of these stuff? (3) For example, consider f: C^2 -> C^2 given by f(z,w) = z^2 + w^2. Then f^{-1}(0) is the paraboloid (i guess). What information I can get about f from this paraboloid? What is the use of this information.

Algebraic geometry is particularly important in number theory, where geometric methods are used in the proof of many basic results (including Fermat's last theorem). In a more practical vein, elliptic curves are the basis of new approaches to primality testing and encryption. Curves over finite fields are also important in coding theory. In a moe general way, you can think about algebraic geometry as providing a method for studying solutions of polynomial equations in situations where it may be impossible or impractical to actually solve them. Greg Woodhouse 23:06, 29 November 2006 (UTC)

I'm not so sure that C* algebras need to be mentioned here, but the ring spectrum is absolutely fundamental. It is nothing more than than the Zariski topology for an affine "piece" of a variety or scheme. Or, if you like, it provides a dictionary for translating algebra into geometry and vice versa. It deserves a prominent place in the article. Greg Woodhouse 23:14, 29 November 2006 (UTC)

Thanks a lot for your time! 201.50.255.41 23:57, 20 March 2006 (UTC)

In the first paragraph I changed algebraic equations to polynomials. Much of algebraic geometry is centered around the study of finitely generated algebras over an algebraically closed field. Thus, rational polynomials would have been too restrictive.--Exoriat 07:04, 6 July 2006 (UTC)

What is confusing is that the term "geometry" has a somewhat different significance here than it does in differential geometry. for example, curvature is an important concept in diferential geometry, but the conept plays no role in agebraic gemetry (because it is not invariant under regular maps). However, such concepts as intersection multiplicity are meaningful in algebraic geometry and do play an important role. It is useful to think about the difference between real and complex analysis. A (complex) analytic function defined in a small neighborhood can be "continued" to a meromorphic function (that is, a function having no essential singularities) in a unique way. the upshot of this is that analytic functions are very rigid, being entirely deterrmined by their values in an arbitrarily small neighborhood. The same is true of regular functions on varieties. The geometry of algebraic varieties is, of necessity, somewhat more global in nature. Locally, it is possible to talk about the degree of a zero (or the intersection multiplicity of two varieties), but that's about all. A good example of a geometric result in Bezout's theorem. which gives the total number of intersection points (counting multiplicity) of two plane algebraic curves as the product of the degrees. A deeper result, the Riemann-Roch theorem tells us that therer is an invariant of a crve known as its genus that is intimately connected with the dimnension of the space of functions that are regular away of from a finite set of points (with multiplicity) known as a divisor. In particular, it identifies a special divisor class called canonical (basically, zeroes of differential) such that l(D) = l(W - D) - deg D + 1, where g is the genus, W is a canonical divisor, the degree of a divisor is the number of points counting multiplicity, and l(D) is the dimension of the space of functions regular outside D. From a modern point of view, this result expresses the duality between H0 and H1 in the cohomology of the sheaf of regular functions on the scheme/variety (in this case, curve). The term genus may seem peculiar, but over the complex numbers, algebraic curves are essentially Riemann surfaces, and the genus is then nothing more than the topological genus. Greg Woodhouse 01:26, 30 November 2006 (UTC)

parabola - smoothness
As it turns out, V(y - x3) has a singularity at one of those extra points, but V(y - x2) is smooth.

I removed this, because y=x^2 is not smooth at infinity (the homogeneous equation is yz=x^2, at z=0 this gives x^2=0, which is not smooth at x=0, therefore the point [0:1:0] is not a smooth point of the projective variety. Jakob.scholbach 04:24, 21 April 2007 (UTC)

Why is the parabola not smooth? Let us simply look at the projective equation. If we take x partial we get 2x, y partial is z, and the z partial is y. Plugging in the point [0:1:0] gives that the z partial is nonzero, and hence it is smooth. Recall that a variety given by a single equation (or more generally a complete intersection given by a system of defining equations with the "correct properties") is singular at a point if and only if all partials vanish at this point (or for a complete intersection the matrix of derivatives is singular). The mistake is plugging in z=0. It's no fair to plug in z=0 and then ignore y. You must first look at an affine piece, given usually by plugging in 1 for one of the variables. Plugging in x=1, gives yz=1 which is smooth (z partial being y and y partial being z. these vanish simultaneously for z=0,y=0, but this is not on the curve yz=1); plugging in y=1 gives z=x^2 which is smooth (z partial is 1); and finally z=1 gives y=x^2 which is smooth (y partial is 1). Gmichaelguy 00:36, 25 April 2007 (UTC)

Thank you. I must have been blind. I revert my mistake. Jakob.scholbach 01:16, 25 April 2007 (UTC)

Dieudonné reference
I moved the following here, since it did not belong where it was in the history section. It was inserted by an anonymous editor, who apparently was not the original author of the section in dispute. Nevertheless, it may be worth using it as an actual reference at some point (particularly in the rather thin later sections on history):


 * See e.g. Dieudonné, Jean: "The historical development of algebraic geometry", Amer. Math. Monthly 79 (1972), 827--866. (MR46#7232) or his more complete "History of algebraic geometry. An outline of the history and development of algebraic geometry", Wadsworth Mathematics Series. Wadsworth International Group, Belmont, Calif., 1985. 186 pp. ISBN: 0-534-03723-2 (MR86h:01004) —Preceding unsigned comment added by 70.20.97.103 (talk) 14:11, 19 November 2007 (UTC)

van der waerden
Could the 'van der waarden' mentioned in the notes and history section of this page be 'van der waerden', who has an article on wikipedia already? —The preceding unsigned comment was added by 80.229.247.11 (talk) 17:16, 7 May 2007 (UTC).

hahah!! goodluck —Preceding unsigned comment added by 125.60.241.39 (talk) 23:28, 31 August 2007 (UTC)

I don't understand your question. I see van der Waerden only once at the page. Jakob.scholbach 09:39, 1 September 2007 (UTC)

Ah...I suspect the page has been fixed since my original comment. Thanks. —Preceding unsigned comment added by 80.229.247.11 (talk) 01:15, 6 April 2008 (UTC)

visualization
I tried to visualize the example of projective closure of the parabola and y=x3 mentioned in the text: it looks like this

Do you think it is OK, or is too messy? —The preceding unsigned comment was added by Jakob.scholbach (talk • contribs) 04:43, 4 May 2007 (UTC).

Hey Jakob, It's nice that you're interested in this. Actually, you will be able to connect this question of what a parabola/ellipse/hyperbola looks like in projective space with the notion of conic sections. The parabola y=x2 can be viewed as the slice when z=1 of the cone yz=x2, it is fun to see why this is a cone. Each line in the cone, through the vertex, is a point in projective space. Algebraically it has genus zero. Your other example y=x3 corresponds to yz2=x3. It is a degree three curve in the projective plane. It has genus zero. I think it can be proven that the genus g is related to degree d by the formula d2-3d=2g-2.

Noproxy (talk) 22:18, 8 January 2009 (UTC)

regular maps vs. regular functions
In this article, the term "regular function" can mean maps from a variety to any other variety. However, the article on regular functions restricts itself only to functions into the ground field. On the other hand, there is an article on regular maps, which describes the meaning of "regular function" of this article. This mix of notation should be changed: Either the references of "regular function" in this article should be replaced by "regular map", or the article on regular functions needs to be changed (and merged with the article on regular maps). --Jarauh (talk) 14:28, 24 September 2009 (UTC)

disputed history
I see from the history page that User:Jagged_85 had made some ridiculous claims that Arabs invented most of algebraic geometry, subsequently prompted the dispute tag by 128.118.24.213. Most of the dubious claims are now deleted by User:R.e.b..

Anyway, I think the history section would be much better starting with Newton's introduction of Cartesian coordinate system in its present form to be the point when algebraic geometry started as the systematic study of zeros of algebraic equations. The reference to Arab's reinvention for solving the cubic (the method is known to Archimedes, more than 1000 years before Khayyam) would be best left to cubic equation. Kommodorekerz 15:04, 25 December 2006 (UTC)


 * While much can be said about the history of mathematics in the Middle East, it is important to bear in mind that this needs to be relevant to the article. It also should be represented in proportion to its WP:WEIGHT.  Discussions of trigonometry and other trivia really have no place in a history of algebraic geometry, unless there is a clear (and unoriginal) way to connect these up with the study of algebraic curves and surfaces.  Also, implying that these developments were somehow relevant to the subsequent creation of algebraic geometry is a clear case of WP:SYNTH unless someone can provide a reference to that effect.  siℓℓy rabbit  (  talk  ) 15:01, 15 August 2008 (UTC)


 * The disputed tag has been in place for a while now but it seems unclear what, specifically, is being disputed. If everything has now been resolved by now then please remove the tag, otherwise it would help to clarify which statements are controversial.--RDBury (talk) 11:21, 24 November 2009 (UTC)


 * Looks like the tag was removed, thanks.--RDBury (talk) 08:08, 26 November 2009 (UTC)

Vs. Geometric algebra?
Could someone explain the difference between algebraic geometry and geometric algebra? The two pages seem to cover different material. With such similar names, they should either be the same page or it should be clear how they differ. —Ben FrantzDale (talk) 01:42, 27 January 2009 (UTC)


 * No, they should not be the same page. The two topics are as different as, say, real analysis and analysis of variance.   siℓℓy rabbit  (  talk  ) 02:53, 27 January 2009 (UTC)
 * There should probably be a "not to be confused with" tag on this page then, shouldn't there? They are clearly different topics though. In math it seems to matter a lot whether a word is used as an adjective or a noun.--75.80.43.80 (talk) 11:22, 26 April 2011 (UTC)

Improving the page
I have just rewritten a large part of algebraic geometry. One of the aims of my edits was to better covering the various sub areas of modern algebraic geometry: except for the scheme theory and its generalizations, none of them were even mentioned in the previous version. My guideline for my edits was Did you know that algebraic geometry is not as esoteric as it could seem?

Reviews and comments would be of great help to improve the result.

I need also some specific help to finish the work. Here is a list of the main points on which I need more specific help:
 * List of subareas: I have omitted to mention algebraic group theory and certainly other subfields. Thus the list of subareas in the lead has to be completed.
 * Each sub area of algebraic geometry deserve a section, especially, the relations between algebraic geometry and number theory, but probably also algebraic group theory and singularity theory. I have not the competence for writing them.
 * Section Modern viewpoint renamed Abstract modern viewpoint: this esoteric section is of no help for everybody. In fact most readers, even those like me which have some knowledge of algebraic geometry, have never heard the name of most theories which are cited, and can understand nothing in this section. The few other ones know the subject and do not get new information. Thus this section has to be rewritten for explaining why these theories have been introduced, which problems of general interest they have solved and why it may be interesting to learn them. If this may not been explained, these theories have not their place here but in a new article Extensions of scheme theory. I am not competent enough to rewrite this section in this spirit and thus need help for that.
 * Section Applications is a biased stub: Clearly a stub. Biased because it does not mention the applications of algebraic groups in theoretical physics not the use of algebraic geometry in public-key cryptography nor in algebraic cryptanalysis. I have not a clear view of what has to be done. A possibility is simply to suppress this section and replace it by sections Algebraic geometry and cryptography and ''Algebraic geometry and theoretical physics. This has to be discussed.

Good new year 2012 D.Lazard (talk) 12:28, 29 December 2011 (UTC)


 * The lead seems to be much too long. The usual rule of thumb is three paragraphs (maybe four for a top level article like this).  Starting with the third paragraph, it seems to be much too detailed.  Ideally, these nine mini-paragraphs need to be combined and condensed into one or two paragraphs (possibly on the history of algebraic geometry and the abstract approach a la Grothendieck which, like it or not, is central to a modern understanding of the subject).   Sławomir Biały  (talk) 13:02, 29 December 2011 (UTC)


 * Some comments, mainly about the lead (disclaimer: I am not an expert):
 * as Sławomir suggests, perhaps it would be better to make it a bit shorter. I do realise that there is a contradiction between this suggestion and the following ones.
 * One thing missing in the lead (though I am not sure what is a good way to state it) is that a.g. studies geometric objects (varieties) via the algebra of polynomials on the variety, and that geometric questions are translated to questions about this algebra, and then studied by alg. methods. If you can state this eloquently, it could replace the first sentence of the lead. Perhaps the Nullstellensatz could be mentioned in the lead.
 * algebraic number theory: I would mention that a.n.t. considers prime numbers as points of a variety, that this allowed to bring alg-geom methods to number theory, and that this could be formally done following the introduction of schemes by Grothendieck.
 * Sasha (talk) 16:20, 29 December 2011 (UTC)

What's a point?
It was my impression that a "point" can correspond to a more general object than a subvariety. Tkuvho (talk) 17:14, 29 December 2011 (UTC)


 * May be in the whole generality and certainly for schemes which do not correspond to a classical variety. But, if I is an ideal in R=K[x1, ..., xn], with K algebraically closed, the points of the variety defined by I correspond, by Nullstellensatz, to the maximal ideals of R/I, while the points of the topological space Spec(R/I) are by definition the prime ideals, i.e. the ideals which define a variety. An example of this view of the points is the scheme definition of generic point (by the way, one of the numerous pages of algebraic geometry which deserve to be rewritten).
 * Thus, even if the discussed sentence is not correct in the whole generality, it is correct from the point of view of classical algebraic geometry, and may be helpful for people which know a little of classical algebraic geometry and want to know what is a scheme. If you have an idea to fulfill this aim with a strictly correct sentence, it would be welcome. D.Lazard (talk) 18:07, 29 December 2011 (UTC)

Improving sections
I have just rewritten the section on projective varieties, with the aim, as for previous sections, to be understandable by users who know a little of algebra and geometry, but nothing else.

I am not so happy with the wikilinks: First, I had to edit several of them to get relevant links (for example, Riemann-Roch theorem for algebraic curves did not mention the relation between the genus and the number of singularities). Secondly, the level of abstraction is frequently to high: I have tagged Riemann-roch page as jargon, but many other pages on algebraic geometry start on scheme or bundle theory without sufficient details on classical algebraic geometry. Another example of an irrelevant mathematical level, probably the worst, is Homogeneous polynomial, where it is not even said that the function defined by a homogeneous polynomial is a homogeneous function, but after only 3 lines on the subject of the article one starts to talk on tensors, which is a different subject of higher level of abstraction. In other cases several very close notions are defined in separate pages, which deserve to be merged. Any ideas to solve this general problem on pages related to algebraic geometry?

In my edits, I have not yet developed the relation between affine and projective varieties (projective closure and affine variety as the complement of a hyperplanes in a projective variety). I have not found any wikipage on this subject. Thus a section on it is probably needed, before to expand it in a separate page? Any opinion or comment?

Any comment on the level of technicality of my edits of this section and the previous ones in "Basis notion"? Should we change "Basis notions" to "Basic notions"?

D.Lazard (talk) 15:45, 3 January 2012 (UTC)


 * A quick comment on affine/projective business: it would be helpful to point out that an affine curve has a unique smooth completion though this is not always obtained by closure in the projective space. Tkuvho (talk) 19:03, 3 January 2012 (UTC)
 * I do not well understand. Are you speaking of real curves or of complex curves? If a real curve has two points at infinity, like an hyperbola, I think that there are 3 smooth completions obtained by the 3 ways of connecting the branches together (two of the completions are connected, while the other is not). On the other hand, if a complex curve has a cusp at infinity, like y-x^3=0, I think that the monodromy is kept by any completion and thus that there is no smooth completion. I am wrong? D.Lazard (talk) 21:12, 3 January 2012 (UTC)
 * The projective completion in a given projective space is not intrinsic. Abstractly, y-x^3=0 is just the affine line, so its (unique) regular projective completion is the projective line. Regards, Liu (talk) 23:17, 3 January 2012 (UTC)


 * Certainly, but as I made it clear in my remark, I was not talking about the projective completion. The smooth completion is certainly "intrinsic" as it is unique.  Tkuvho (talk) 10:12, 4 January 2012 (UTC)


 * As regards smooth completion, at least over an algebraically closed field every curve has a unique smooth birational model. You can construct it by taking the normalization of the curve.  Its points correspond to the discrete valuations of the function field.  You can construct it more explicitly by taking any completion, then blowing up the singularities.
 * I'm not quite sure what happens over a non-algebraically closed field. My intuition is that every embedding of the function field into an algebraically closed field determines a normalization and therefore a smooth model.  But such a model is always irreducible, so it would miss the disconnected completion in your example.  I don't know how one would algebraically construct your disconnected hyperbola example.  Ozob (talk) 00:11, 4 January 2012 (UTC)
 * I understand where is the misunderstanding: First I was given to "smooth" the differential meaning, the only one which is defined in wikipedia. This should be corrected by defining smooth variety in singular point of an algebraic variety and editing the dab page of smooth. Secondly I was given to "completion" the meaning of compactificaion or, at least of embedding in a larger topological space. It seems that the result that Tkuvho had in mind and that Ozob refer to is the following:
 * For any affine algebraic curve V there is a unique non singular algebraic curve W and a surjective regular map from W onto V, which is a birational equivalence and is minimal in the sense that it may not be factored through a "smaller" surjection. The curve W is the normalization of V and is defined as the curve which has as coordinate ring the integral closure of the one of V.
 * I agree to mention this at the end of Section "rational function and birational equivalence", but I have a problem for providing a correct wikilink (for such a general page, a link to a specialized page seems better than an external source): The better link I have found is example section of Dedekind domain. Better link?
 * D.Lazard (talk) 10:37, 4 January 2012 (UTC)


 * Thanks, that's excellent (to answer your question about the link: I am not sure what a better link would be). It could perhaps also be mentioned that the smooth completion is uniquely determined by the field of rational functions on the curve.  I have a rather summary understanding of this so I hope I didn't get any details wrong.  Tkuvho (talk) 10:45, 4 January 2012 (UTC)


 * The completion is usually understood as in Ozob's comments: it is a projective curve which contains the given curve as an everywhere dense open subset, and the added new points are regular points (smooth points if the base field is algebraically closed or more generally perfect). Such a (projective) completion always exists and is unique. If the base field is not perfect, a smooth (projective) completion doesn't always exist, even if we start with a smooth affine curve. As for the normalization, over an imperfect field, the normalization is not always smooth. Liu (talk) 21:34, 4 January 2012 (UTC)


 * Let me see if I understand correctly what you're saying. Normal = R1 + S2, so a normal curve is regular.  Over an imperfect field, being smooth over the ground field is not the same as being a regular ring.  If I understand you correctly, you're saying that there are regular curves which are not smooth, and that normalization does nothing to make a non-regular curve any smoother.  So the following would be a counterexample: Let k be a field of characteristic p, K = k(tp), L = k(t), and R = P$1 L$.  R is regular (and normal), but not smooth over K, right?  This is what you had in mind?  Ozob (talk) 02:10, 5 January 2012 (UTC)
 * Yes. Though in this case, there is no a single smooth point in the regular curve. If you consider the (regular) affine curve $$y^p=(x^p-t)x$$, it is smooth except at $$(x^p=t, y=0)$$. The smooth part of this affine curve has no smooth compactification. Of course, I think in this article we shoud not talk about imperfect fields. Liu (talk) 09:14, 5 January 2012 (UTC)
 * OK, thanks! Ozob (talk) 11:13, 5 January 2012 (UTC)
 * Please start a page smooth completion of a curve or smooth projective completion of a curve or smooth (projective) completion of a curve. Tkuvho (talk) 12:30, 5 January 2012 (UTC)
 * I started a stub Smooth completion based on Ozob's comments. It needs to be improved and possibly redirected.  Tkuvho (talk) 18:14, 5 January 2012 (UTC)

Alfred Clebsch
Does Alfred Clebsch deserve mention in the 19th-century history section? Some writers call his work "the birth of modern algebraic geometry". Jamesdowallen (talk) 07:07, 18 April 2012 (UTC)

Should the Stack Project be a reference
?? — Preceding unsigned comment added by 18.111.109.122 (talk) 00:31, 15 December 2013 (UTC)


 * The Stacks Project would make a good external reference. I've added it to the article.  Ozob (talk) 17:58, 15 December 2013 (UTC)

New lead
I have reverted a recent edit of the lead, because it is WP:original synthesis and introduces blatant mistakes. Indeed the new first sentence was :"Algebraic geometry is a branch of mathematics that studies polynomial equations using geometry." It is exactly the contrary: "algebraic geometry" studies geometry using polynomial equations.The second sentence starts as "Classically, this involves real solutions that are usually zero", which is nonsense. Later on, conic sections are omitted as basic staples, although they are much more important than cubic and quartics curves. I could continue to analyze this new lead, but this seems sufficient to show that it cannot be accepted. D.Lazard (talk) 16:33, 24 May 2014 (UTC)
 * Don't bother with it any more. Having just revisited it, I agree that it fell off the rails. I must've been too distracted. Apologies, Sardanaphalus (talk) 22:44, 24 May 2014 (UTC)

Disambiguation request
Algebraic geometry is also the titles of many books, including the famous one by Hartshorne. We need to make Algebraic geometry (disambiguation) page. --Acepectif 20:15, 12 October 2006 (UTC)
 * Any reasonably large area of science will also be the title of many textbooks given the usual, rather unimaginitive, naming conventions. That can't be a reason that a disamb page is needed for all of them. —Preceding unsigned comment added by 90.229.231.115 (talk) 22:58, 6 December 2007 (UTC)


 * algebraic geometry = universal algebra + logic + fields QuentinUK (talk) 11:40, 8 March 2015 (UTC)

Assessment comment
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Changing Introduction
I propose the introductory portion of the algebraic geometry section should introduce scheme theory, use it as a tool for explaining classical theories of varieties all while providing motivation for arithmetic geometry. This can be done through looking at the projective fermat scheme
 * $$X=\text{Proj}\left( \frac{\mathbb{Z}[x,y,z]}{(x^n + y^n - z^n)}\right)$$

and considering various sets, such as $$X(\mathbb{Q}),X(\overline{\mathbb{Q}}), X(\mathbb{C}), X(\mathbb{F}_q)$$. — Preceding unsigned comment added by 128.138.65.70 (talk) 22:37, 10 July 2017 (UTC)
 * Please, don't forget to sign by four tildes ( ~ your post in talk pages.
 * I have reverted your edit for the following reasons:
 * It does not respect the guidelines of MOS:LEAD, which asserts (among other rules) "The lead should stand on its own as a concise overview of the article's topic. It should identify the topic, establish context, ..., and summarize the most important points".
 * It is far too WP:TECHNICAL: the lead must be accessible for readers who do not know anything of the subject. Your lead begins with the use of terminology and notation (Proj), which can be understood only by people who know already schemes.
 * It does not respect WP:NPOV, by asserting that modern algebraic geometry reduces to scheme theory ("Schemes subsume the classical theory of algebraic varieties"). This point of view is blatantly wrong, as there are many active areas of algebraic geometry for which schemes are not useful and not used. Practitioners of these areas of algebraic geometry often ignore what is a scheme. Such parts of algebraic geometry include real algebraic geometry, computational algebraic geometry, singularity theory, elliptic and hyperelliptic curves over a finite field (widely studied for cryptography), etc.
 * I could continue to enumerate the other issues of this edit, but these points are sufficient for showing that it cannot be accepted. D.Lazard (talk) 08:38, 11 July 2017 (UTC)


 * I'm with D. Lazard on the change. I also want to point the lead does introduce the scheme theory, which is a kind of a general version of Hilbert nullstellensatz. I quote "This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic...". I think this description is more accessible to those who don't have background in the scheme theory. Having said this, it's perfectly acceptable, I think, to have some technical discussions (e.g., rational point in the scheme theory) in the main body of the text. -- Taku (talk) 17:46, 11 July 2017 (UTC)

Further Reading Section Booklist
Hey guys, I am new here so please correct my etiquette if I say something wrong. This is a minor suggestion, but I think Ravi Vakil's book/lecture notes "Foundations of Algebraic Geometry" should be on the subsection "Textbook and reference for schemes". It is very self-contained, and personally I think it is both a great (but hard) introduction to Modern Algebraic Geometry.Leon2k2k2k (talk) 01:33, 21 July 2017 (UTC)