Talk:Algebraic geometry

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Mathematics grading:	B Class	High Importance	Field: Geometry and topology
Needs gentler introudction to subject, pictures, and more examples. Potential Good Article nominee. Tompw 19:05, 5 October 2006 (UTC)

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 x² + y² + z² − 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)

[edit] 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)

[edit] 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)

[edit] 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)