Talk:Algebraic curve

Is this standard terminology?[edit]

The article states:

"A nonsingular n-dimensional complex projective algebraic curve will then be a smooth orientable surface as a real manifold, embedded in a compact real manifold of dimension 2n which is CP^n regarded as a real manifold."

Is it standard terminology to call a projective variety of any dimension a "curve" ??? I had believed that this terminology was reserved for projective varieties of (complex) dimension 1. Daqu (talk) 06:43, 19 March 2008 (UTC)[reply]

Certainly not. The "n-dimensional" is simply not appropriate here, so I removed it. Joerg Winkelmann (talk) 01:40, 20 August 2008 (UTC)[reply]

Singularities[edit]

This section has a passage that reads:

If, for example, we simply look at a curve in the real affine plane there might be singular P modulo the stalk, or alternatively as the sum of m(m−1)/2, where m is the multiplicity, over all infinitely near singular points Q lying over the singular point P. Intuitively, a singular point with delta invariant δ concentrates δ ordinary double points at P.

It seems as though some text has been deleted here, but I can't find the missing piece in the history. Maybe the original author could restore it. Artie P.S. (talk) 08:33, 20 April 2009 (UTC)[reply]

I don't understand what this phrase is supposed to say. The guys means that if you have an affine curve, which is smooth, and then you take the closure in some projective space, then singularities may appear?

Certainly. In practical terms this is quite likely to happen. Charles Matthews (talk) 11:11, 17 September 2009 (UTC)[reply]

Since when is an algebraic curve necessarily a surface???[edit]

The article reads:

"An algebraic curve likewise has topological dimension two; in other words, it is a surface."

A surface is by definition a manifold. Algebraic curves may have singularities that prevent them from being everywhere locally Euclidean. Therefore it is false to state without further conditions that an algebraic curve "is a surface".

Can we perhaps limit the article to true statements only?Daqu (talk) 08:19, 22 November 2009 (UTC)[reply]

Yes, add "away from its singular points". Charles Matthews (talk) 08:39, 22 November 2009 (UTC)[reply]

In my opinion, that's way too vague. Anything is a manifold "away from its singular points." (Especially if you define "singular point" as "where the thing is not a manifold".) Daqu (talk) 01:04, 23 November 2009 (UTC)[reply]

Not vague if as [[singular point of an algebraic variety|singular point]]. Charles Matthews (talk) 09:20, 23 November 2009 (UTC)[reply]

Of course, you are absolutely right, needless to say. The only reason I'm harping on this issue is that, as little as I know about algebraic curves (very little!), I do know that "resolution of singularities" is a fundamental area of study concerning algebraic varieties in general. So it would not be appropriate to give the misimpression that the singular points are just some negligible part of an algebraic curve, the manifold part being all that matters. (Even if the set of singular points is always of lower dimension than the rest of the curve.)Daqu (talk) 02:05, 6 December 2009 (UTC)[reply]

Definition and context[edit]

The definition given in terms of algebraic varieties is technically true, but I believe the average person coming to this article will not be familiar with the terminology of algebraic geometry. In fact I would think that this article should lay a foundation from which the reader would be able to better understand the general theory. The lead section should be rewritten to be accessible to familiar with freshman calculus and sections to introduce the terminology should be added at the beginning of the article. A history section would be helpful as well.--RDBury (talk) 13:34, 6 January 2010 (UTC)[reply]

Also the context is not historically accurate. Algebraic curves were already studied as far back as the 17th century (and probably earlier), so maybe change it to "quite fully developed by the 19th century". But even this statement is not satisfactory, major advances in the theory of algebraic curves were also accomplished in the 20th century as well. Wishcow (talk) 08:56, 10 October 2011 (UTC)[reply]

Plane curve[edit]

I very much doubt what is said about plane algebraic curves. In the example with the curve determined by

4x^{2}+y^{2}-z^{2}=1

and

z=x^{2}

we may eliminate z (in one of the equations) yielding $4x^{2}+y^{2}-x^{4}=1$ , but although this determines a curve in the xy-plane, there still is the z-dependance. For $x=0$ , $y=\pm 1$ and $z=0$ , for $x={\tfrac {1}{2}}$ , $y=\pm {\tfrac {1}{4}}$ and $z={\tfrac {1}{4}}$ , etc. Nijdam (talk) 21:15, 23 March 2012 (UTC)[reply]

Possible article structure[edit]

I hope to work on the article sometime soon, so here is a draft of its structure.

Motivation and Definition: start with F(x,y)=0.
- Examples: A^1, P^1, rational curves, plane curves (Bézout's theorem), affine
- (resolution of) singularities
Classical alg. geometry : curves over alg. closed fields. are described by their function field, smooth completion, rational points,
Points = Divisors: Riemann–Roch theorem, Abel–Jacobi map, Jacobian variety
Curves over C and Riemann surfaces:
Moduli: (semi)stable curves
Arithmetic: Fermat curve?, Modularity theorem, L-functions, modular curve/form, Mordell–Weil theorem, Faltings' theorem, Counting points on elliptic curves, Weil conjectures, height (pairing)?, BSD conjecture, dessin d'enfant
Applications: Elliptic curve cryptography, Elliptic curve primality testing, Elliptic curve primality proving

Jakob.scholbach (talk) 12:15, 26 September 2012 (UTC)[reply]

I agree with your project. But it seems that you (and also the authors of the present version) do not take into account that most readers of this article have not a graduate level, and thus are not familiar with algebraic geometry. Moreover there are probably interested mainly in the real points of a plane algebraic curve defined by an implicit equation. This could be the subject of an article in WP, but, presently, it is not. Thus, IMO, a first section has to be added to your project, which could be named "Plane affine curves and their real points". This section could contain

Intersection with lines and other curves, Bézout's theorem, how to compute the intersection, link to Resultant
Tangent at a point
Inflexion points, Hessian
Asymptotes. How to compute them algebraically
Singularities. How recognize their type algebraically. How to compute them (system of polynomial equations or multiple roots of the discriminant w.r.t. one of the variables)
Critical points in a direction, link to Discriminant
Determination of the topology of the real points
Parametric curves, implicitization, ...

It is also possible to create an article Plane algebraic curve or Implicit curve (presently, both redirect here) and to have a short section with template {{main}}. IMO, developing these points, which are of interest for a rather large public (including college students) is much more important than improving things for a very narrow audience. — D.Lazard (talk) 13:28, 26 September 2012 (UTC)[reply]

Yes, the article should start out gently and real curves is a good first topic in this sense. However, it is probable that not all of the aspects you mention can be covered in detail due to length restrictions. Jakob.scholbach (talk) 08:04, 27 September 2012 (UTC)[reply]

When the article will become too long, it will be the time to split it and use the template {{main}} and/or multiple cross links. More, some of my items would deserve a separate article like "topology of algebraic curves" or "implicitization" (of curves and surfaces). D.Lazard (talk) 10:02, 27 September 2012 (UTC)[reply]

New lead[edit]

I have expanded the lead in order to take into account the various meanings of "algebraic curve" and to make the lead understandable to the reader that knows nothing of algebraic geometry. The problem was that there are several points of view on the subject. The point of view of Euclidean geometry was completely omitted by the previous version of the article. Splitting the article in an article about the point of view of real geometry and an article about the point of view of abstract algebraic geometry may be needed in the future, but is impossible with the present state of the article, because too much basic information is lacking, that is common to the two points of view (for example the algebraic computation of the asymptotes in Euclidean geometry needs some notion of projective space, and Bézout's theorem is fundamental in Euclidean geometry, but may be well understood only in complex geometry).

With the new lead, the article needs clearly to be expanded in the lines sketched in the preceding section of this talk page. --D.Lazard (talk) 14:16, 24 October 2012 (UTC)[reply]

contradiction?[edit]

What does this mean (from section 'intersection with a line')?

1 = q(y) = 0 (or 1 = q(x) = 0) 173.25.54.191 (talk) 02:48, 2 May 2014 (UTC)[reply]

The "1 =" is probably the mistaken result of the suppression of a template {{math}} or {{nowiki}}. I have restored the correct formulas ans added a minor clarification. D.Lazard (talk) 09:10, 2 May 2014 (UTC)[reply]

What does this mean?[edit]

In the projective plane section, there's this:

^hP(x,y,1)=P(x,y,z), as soon as the homogeneous polynomial P is not divisible by z.

I'm good with P(x, y, z) = 0 reduces to P(x/z, y/z, 1) = 0 (by dividing out z^k) when z isn't 0, but I can't make that statement make sense.

173.25.54.191 (talk) 01:17, 5 May 2014 (UTC)[reply]

I agree that then notation is unclear. I'll try to correct it. D.Lazard (talk) 11:28, 5 May 2014 (UTC)[reply]

Plücker formulas[edit]

Would a brief mention of the Plücker formulas and a citation of their Wikipedia page be appropriate here? Ishboyfay (talk) 00:12, 7 December 2014 (UTC)[reply]

Yes, it will be appropriate. However, it is not clear to me where and how to make this mention. Maybe in a new subsection of section "Example of curves", entitled "Dual of a curve". Maybe you could write it? D.Lazard (talk) 10:10, 7 December 2014 (UTC)[reply]

Closed curve[edit]

Given the polynomial f(x, y) and the implicit equation f(x, y) = 0, is there a way to determine whether its curve is closed? (If so, maybe this should go into the article.) Loraof (talk) 20:18, 28 September 2017 (UTC)[reply]

An algebraic curve (and more generally an algebraic set) is always closed, being the inverse image of a closed sed {0} by a continuous map (a polynomial or a vector of polynomials). More interesting is the question whether a real algebraic curve is compact. It is compact if it has no real asymptotic direction, that is if the homogeneous part of highest degree of its equation has no real 0 (except (0, 0)). However, a compact curve may have real asymptote directions, corresponding complex conjugate branches having the same real asymptote. More precisely a real algebraic curve is compact if and only if all real points at infinity are isolated (that is all their tangents are non-real). This may be tested by computing the real points at infinity. The curve is compact, if and only if all real points at infinity have an even multiplicity and a tangent cone, which is an union of non-real lines. D.Lazard (talk) 21:50, 28 September 2017 (UTC)[reply]

Sorry, I may not have expressed myself right. The article Curve says A closed curve is a curve that forms a path whose starting point is also its ending point—that is, a path from any of its points to the same point. So for example an ellipse is closed in the real plane but a parabola or hyperbola is not. The article Conic section#Discriminant shows how to tell, in terms of the polynomial's parameters, whether a quadratic in two variables is an ellipse or not. Is there any way to tell for a polynomial equation of arbitrary degree whether a path in the real plane from any point traverses the entire curve and leads back to that original point? Loraof (talk) 22:32, 28 September 2017 (UTC)[reply]

During the night (that is before reading this post), I have understood this confusion. However, the definition that you have quoted applies only on curves that are connected manifolds. Should we consider an eight curve as a closed curve? Should we consider a non-connected Cassini oval a closed curve? In fact, my above comment on compactness is a partial answer to your question, as a closed curve in the sense of Curve is necessarily compact.

In fact, your question has to be generalized into two questions that deserve clearly to be considered in this article: 1/ What can be the topology of a plane algebraic curve? 2/ Can the topology of an algebraic curve be completely determined? The answer of the second question is yes. However, there is still active research on this question, because of the difficulty designing efficient and robust (that is certifying correctness of the result) algorithms. The complete answer to the first problem is still an open problem, as it includes Hilbert's sixteenth problem. Nevertheless, the article should include the following results that are a useful answer to the question 1:

Local topology: In a sufficiently small (open) neighborhood of one of its point a plane algebraic curve is either reduced to this point (isolated point) or is the union of a finite number of branches that are diffeomorphic to a line or a cusp. This implies that if one enters into a singularity by following a branch of an algebraic curve, the branch by which one should go out is fully determined, or in other words that an algebraic curve is the union of a finite number of (possibly self-crossing) topological curves without end points.

Global topology: An algebraic plane curve is a finite disjoint union of isolated points (if any), open smooth monotone arcs, singular points and regular points with horizontal or vertical tangents, such that the endpoints of monotone arcs are either at infinity or one of the preceding specific points. Moreover, there is an even number of arcs that have a given point as an end point, and this number is 2 for the regular points with horizontal or vertical tangent. Thus, computing the topology amounts essentially to compute the remarkable points, points on each smooth arc and relationships between points (including points at infinity) induced by the arcs.

By the way, I have published, several years ago, a case of study, which contains the study of an algebraic curve of high degree, and illustrates some strange properties that an algebraic curve may have, the ability of modern technology to compute the topology, and the difficulty of plotting an algebraic curve of high degree, even if it comes from an application.^[1] D.Lazard (talk) 08:38, 29 September 2017 (UTC)[reply]

I have added a section Algebraic curve § Analytic structure, where the above local topological properties are explained with the aid of Puiseux series. Although the material in this section is well known since Puiseux, I am unable to provide sources. D.Lazard (talk) 21:11, 25 January 2018 (UTC)[reply]

References

^ Lazard, D. (2004). Injectivity of real rational mappings: the case of a mixture of two Gaussian laws. Mathematics and Computers in Simulation, 67(1), 67-84.

Examples[edit]

There should be a discussion of complete intersection curves along with their genera, seen here https://mathoverflow.net/questions/114483/genus-of-non-complete-intersections along with a complete list of algebraic curves. Mumford gives an excellent list in his book on curves and their jacobians: https://wstein.org/edu/Fall2003/252/references/mumford-jacobians/ — Preceding unsigned comment added by Wundzer (talk • contribs) 23:10, 26 February 2020 (UTC)[reply]

[1] Lazard, D. (2004). Injectivity of real rational mappings: the case of a mixture of two Gaussian laws. Mathematics and Computers in Simulation, 67(1), 67-84.

[1]