Talk:Algebraic curve

message: 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)
thread_title: Is this standard terminology?

message: 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)
replies: text: Certainly. In practical terms this is quite likely to happen. Charles Matthews (talk) 11:11, 17 September 2009 (UTC)
thread_title: Singularities

message: 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)
replies: text: Yes, add "away from its singular points". Charles Matthews (talk) 08:39, 22 November 2009 (UTC)
replies: text: 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)
replies: text: Not vague if as singular point. Charles Matthews (talk) 09:20, 23 November 2009 (UTC)
replies: text: 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)
thread_title:  Since when is an algebraic curve necessarily a surface??? 

message: 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)
thread_title:  Definition and context 

message: I very much doubt what is said about plane algebraic curves. In the example with the curve determined by 
:4 x^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 12, y= \pm \tfrac 14 and z = \tfrac 14, etc. Nijdam (talk) 21:15, 23 March 2012 (UTC)
thread_title:  Plane curve 

message: 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)
thread_title:  New lead 

message: 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)
replies: text: The "1 =" is probably the mistaken result of the suppression of a template  or . I have restored the correct formulas ans added a minor clarification. D.Lazard (talk) 09:10, 2 May 2014 (UTC)
thread_title:  contradiction? 

message: 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)
replies: text: I agree that then notation is unclear. I'll try to correct it. D.Lazard (talk) 11:28, 5 May 2014 (UTC)
thread_title:  What does this mean? 

message: 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)
replies: text: 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)
thread_title:  Plücker formulas 

message: 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)
thread_title:  Examples