Talk:Plane curve

Algebraic curve of degree 2

@D.Lazard: You recently added this to the section Plane curve#Examples:

an algebraic curve of degree less than 3 is always contained in a plane

I don't understand why this is true. For example, what about the curve represented by the two second-degree polynomial equations

$x^{2}+y^{2}+z^{2}=a$

$x^{2}+bxy+cy^{2}+dz^{2}=e$

It's an algebraic curve according to Algebraic curve#Non-plane algebraic curves, it has degree 2, but it is not contained in a plane. Loraof (talk) 19:14, 25 September 2017 (UTC)

To editor Loraof: The intersection of two surfaces of degree 2 is a curve of degree 4, except if the surfaces have a common component (in this case the intersection is not a curve) or if the intersection has a component at infinity. A simple proof of the fact that a curve of degree 2 is a plane curve is the following: the intersection of an irreducible curve of degree 2 with a plane either consists of at most two points, or contains the curve. Thus any plane containing three points of the curve contains the curve, and the curve is a plane curve.

It is funny that you ask this question when I was writing Hilbert series and Hilbert polynomial#Complete intersection, which contains a proof that your example has degree 4. D.Lazard (talk) 20:17, 25 September 2017 (UTC)

Merge with algebraic curve?

Much of the content in algebraic curve on plane curves is duplicated here. Either most of that content should be moved here or this article should be merged into algebraic curve. — MarkH21 (talk) 19:00, 14 April 2019 (UTC)

I have not a clear opinion. We have also curve, which has also a large part devoted to algebraic curves. In fact, there are essentially four cases (plane or not, differentiable or not). The distinction between plane or not is not very meaningful in the differential case, as one generally work with a parametrization. In the algebraic case, a plane curve is defined by a single equation, while, in the non-plane case, it may be difficult to distinguishing between a curve and a variety of higher (or lower) dimension. Nevertheless, the study of non-plane curves passes generally by the study of their projections as plane curves. So, an article on algebraic curves can naturally be structured as a first part on plane curves and a second part on non-plane curves, which uses some results of the first part.

A solution may be to transform curve into a dab page (or a short article explaining the two cases) linking to Differentiable curve and Algebraic curve. This would be a major restructuring, so further advices are needed. D.Lazard (talk) 19:50, 14 April 2019 (UTC)

Ah, I do like that idea. Even before looking at curve, it's strange that the almost the entire lead & the first half of the article on algebraic curve is about plane algebraic curves, whereas this article is very short and stubby. Most of Differentiable curve (a very short section) is really discussed at Differential geometry of curves.

My proposal for the details of the restructuring is:

1. differentiable curve being made into a standalone article combining the smooth plane curve section here, differential geometry of curves, and the relevant short parts of curve, with a section on differentiable plane curves, smooth plane curves, etc.
2. algebraic plane curve being made into a standalone article combining the algebraic plane curve section here and a good chunk of the article on algebraic curve, and the relevant parts of curve,
3. curve being a short list article that links to differentiable curve, algebraic curve, etc.
4. plane curve redirecting to curve

The algebraic curve article can then be shortened and refer to the main article on algebraic plane curves. The existing material on differentiable plane curves seems quite short so it can be a section in the differential curve article for now. — MarkH21 (talk) 04:55, 15 April 2019 (UTC)

— Notified WikiProject Mathematics in case anyone else wants to provide input. — MarkH21 (talk) 05:08, 15 April 2019 (UTC)

If we're talking about how our various articles on curves are to be structured, shouldn't we also include Jordan curve? In any case, we certainly need an article on curves that are not assumed to be algebraic or even smooth (e.g. to support topics like curve-shortening flow and inscribed square problem), so merging this article into an article on algebraic curves seems like a non-starter to me. —David Eppstein (talk) 06:19, 15 April 2019 (UTC)

@David Eppstein: Sure, this doesn't have to be merged or turned into a redirect per se, but the overlapping material can be collected into standalone articles on algebraic plane curves, differentiable plane curves, etc. and this article can have a short description linking to those new main articles. In other words, remove part 4 of the proposal. — MarkH21 (talk) 06:57, 15 April 2019 (UTC)

Too much much redundant material overlaps could be eliminated using redirects to sections in other articles. However seperate terms should have their own entries (as David has pointed out above) and a certain amount of redundancy is actually ok. Excessive redundancy/overlaps should be removed or reduced, but a complete removal is not a good idea. For instance we should not force people looking for a quick oberview on plane curves to read the article on algebraic curves. People looking for quick overview/summary of one term should normally get that from one article and not being forced into link hopping.

So with regard to original question I'm against merging plane curve and algebraic curve and the current amount of redudancy in planar curve are reasonable. However content/aspects not fitting in algebraic curves could be extended. --Kmhkmh (talk) 09:31, 15 April 2019 (UTC)

I agree that "People looking for quick overview/summary of one term should normally get that from one article and not being forced into link hopping". But the present structure is very far to satisfy this. We must not hope that a reader who need information on some kind of curves really knows whether his/her curves are differentiable of algebraic. So we may suppose that the first article that he/she will read is Curve. This article is fundamentally misleading, as the section "Definition" asserts that a curve is a differentiable curve, that the existence of algebraic curves is mentioned only in the last section, and it is not even said that there are curves that are neither differentiable nor algebraic, such as (for example) a plane curve defined by a non-polynomial implicit equation. So, a first task would be to rewrite this article for being usable as a guide among the different sorts of curves. Clearly this article must contain some kind of summaries of Differentiable curve (which should be a redirect to, or a new name for Differential geometry of curves) and Algebraic curve. It must also describe or link to other kinds of curves, if any.

About algebraic curves: It is true that a large part of the article is devoted to the plane case. Some posts suggest that we could have a separate article for this case. IMO, this is not a good idea, as section "Non-plane algebraic curves" shows clearly that everything that can be said about non plane algebraic curve may be deduced from plane curves.

So, my suggestion (possibly different from what I said in my above post) is

• Rewrite Curve as a broad-concept article
• Move Differential geometry of curves to Differentiable curve, leaving a redirect
• Keep Algebraic curve
• Merge or redirect other articles about curves to one of these three articles

This is not very different form MarkH21 proposal. D.Lazard (talk) 11:23, 15 April 2019 (UTC)

I somewhat agree with this suggestion except for the last point. I'm not quite convinced that we should only have those 3 cases and nothing else. A separate summary article plane curves, their most common examples and applications makes at least sense to me. In addition to mathematical aspects of structuring, we also should have in mind accessibility aspects, that is important special cases requiring less domain knowledge may deserve their own article as well, which usually reflect their treatment in literature anyhow.--Kmhkmh (talk) 19:05, 15 April 2019 (UTC)

Certainly space curves, Jordan curves, plane smooth curves, plane real algebraic curves, and algebraic curves as a general concept in fields that might not be the reals are all five different concepts from each other deserving of at least five separate articles. With maybe a sixth general-concept article to disambiguate them. So Lazard's proposal still has too few kinds of curves. —David Eppstein (talk) 05:31, 16 April 2019 (UTC)

I agree that we don't need to combine all articles on curves (i.e. the fourth point of Lazard's proposal). Otherwise I'm on board with the first three. Regarding algebraic non-planar curves, the section here in the article on plane curves discusses only affine curves and only up to bi-rational equivalence. There are significant and diverse fields of mathematics dedicated to algebraic space curves (algebraic non-planar curves), e.g. the enumerative geometry of counting elliptic curves on toric surfaces. The theory of algebraic space curves does not reduce to the theory of algebraic plane curves. On this matter, I would move a lot of the algebraic plane curve material to algebraic plane curve (while leaving an overview at algebraic curve and a link to the new article) and expand the non-planar section of the algebraic curve article which misleadingly suggests that there is nothing of interest with algebraic space curves. — MarkH21 (talk) 05:52, 16 April 2019 (UTC)

Sure, algebraic space curves are interesting, but we should not pretend that they are the only kind of space curves. (Even such basic space curves as a helix are not algebraic.) —David Eppstein (talk) 06:16, 16 April 2019 (UTC)

@David Eppstein: Of course. I'm only talking about algebraic space curves in relation to the restructure of the article algebraic curve and writing algebraic plane curve as a new article. It was in response to everything that can be said about non plane algebraic curve may be deduced from plane curves. — MarkH21 (talk) 06:39, 16 April 2019 (UTC)

Moved discussion from Wikiproject Mathematics

(The following discussion has been moved from the math project talkpage.)

I should also mention that there is a draft article at [1]. As I see, the problem is that algebraic curve is trying to do too much. While the article is fine as it is, it's quite awkward to add more extensive algebraic-geoemtry content to it; e.g., discussions of canonical curves or stable curves. -- Taku (talk) 05:48, 16 April 2019 (UTC)

One possible solution is to have the article projective curve (a projective variety of dimension one). -- Taku (talk) 05:52, 16 April 2019 (UTC)

Absolutely! That's why I suggested moving a large portion of the article to algebraic plane curve while expanding the section on algebraic space curves in algebraic curve. Your suggestion is also a plausible alternative, but that can probably be done independently of this issue with plane/space curves. — MarkH21 (talk) 06:08, 16 April 2019 (UTC)

(Sorry for somehow splitting the discussion but my comment is not strictly about plane curves). I don’t think “algebraic space curves” is a common term but projective curve is. From what I can see, the article algebraic curve is actually mainly about algebraic plane curves or trying to reducing to the plane case. Nothing wrong with that approach itself but that would suggest redirecting algebraic plane curve to algebraic curve is correct. My problem is, with this structure, it’s quite hard to adopt more abstract/intrinsic point of view, which is essential if you’re interested in say enumerative geometry stuff (e.g., degeneration of curves). I think the usual solution here is to just use separate articles that adopts differing point of views. (Unless anyone objects, I think I will just start projective curve.) —- Taku (talk) 10:21, 16 April 2019 (UTC)

Note there is no section on classification of algebraic curves in the algebraic curve article; this is because, in algebraic geometry, one usually classifies projective curves. —- Taku (talk) 10:27, 16 April 2019 (UTC)

Correction: a stable curve is only a complete curve not a projective one. (misleading comment). So complete curve might be a better article title. (Update: there is now Draft:Complete algebraic curve.) By the way, this shows this is an "adjective" problem as MarkH21 has put in a sense: depending on context, an (algebraic) curve can mean a projective one or complete one or smooth projective one (the case in Hartshorne's algebraic geometry), etc. -- Taku (talk) 04:08, 17 April 2019 (UTC)

Sorry, but some assertions in the preceding post and in the draft must be clarified.

• An algebraic space curve is not an old name for a projective or complete curve, as asserted in the draft. It is an algebraic curve that is also a space curve, that is an algebraic curve, or, in other terms, a non-plane algebraic curve.
• When talking of classification of curve, I guess that under birational equivalence should be understood.
• AFIK, every algebraic curve may be completed in a unique way to a complete curve which is also a projective curve, and this completion is compatible with binational equivalences. Moreover, every curve is birationally equivalent with a nonsingular curve (desingularization is easy for curves). So, only nonsingular complete curves need to be considered for the classification under birational equivalence. I may have done minor error in the preceding assertions, but after being fixed, they belong to Algebraic curve.
• AFIK, every plane (affine or projective) algebraic curve may be desingularized by lifting it to a nonplane curve as its inverse image under a projection. Conversely projecting on a plane a non plane curve is often useful for studying it; for example, the singularities of the projection may be used for computing the genus.

So, it is difficult to split the plane and the non-plane cases in a conceptually simple way. So, IMO, we must wait that Algebraic curve becomes too large before discussing for splitting it in sub articles. D.Lazard (talk) 08:28, 17 April 2019 (UTC)

I have removed “algebraic space curve” bit from the draft (it was just meant the term is used less today); for the classification, yes I will add more details on the meaning of equivalences (that is, birational equivalence is isomorphism for smooth complete curves). For the last part, I don’t know if I follow. The draft is not an attempt to separate plane and non-plane cases: in fact, I agree it’s better to discuss the plane case in the algebraic curve article, as the plane case is most familiar and standard one. The draft is however an attempt to adopt more abstract point of view; I think the article is already long enough to warrant a separate article on say complete or projective curve (both essentially the same concept). (When a curve is not embedded, the question on plane or not is meaningless after all.) —— Taku (talk) 09:15, 17 April 2019 (UTC)

That the article is not “too long” is not necessary a strong argument against a separate article; one could, for example, argue “projective variety” is not needed since the topic can be covered at “algebraic variety”. To me, there are enough materials for projective or complete curves as a separate article (by the way, there is no need to separate the complete and projective cases since they are essentially the same). —- Taku (talk) 11:55, 17 April 2019 (UTC)

@D.Lazard: I have put back "space curve" bit to the article, as the term need to be mentioned. I still think we need to mention it's dated term but not sure the best way to do. -- Taku (talk) 23:25, 20 April 2019 (UTC)

@TakuyaMurata: It's not really that dated though. Some examples: Hartshorne 1994, Crelle 1999, RIMS 2006, Annals 2017. — MarkH21 (talk) 09:08, 21 April 2019 (UTC)

@MarkH21: “dated”, I admit, was a probably wrong adjective to use; of course, the term is still in use. What I wanted was to point out standard textbooks much prefer modern terms like “complete curve” or “projective curve”; e.g., Hartshorne, Algebraic Geometry or Geometry of Algebraic Curves by Enrico Arbarello, Joseph Harris, Maurizio Cornalba, and Phillip Griffiths or Moduli of Curves by Joe Harris, Ian Morrison. These textbooks discuss space curves but without the prominent use of the term “space curve”. The term “space curves” still but is often in historical contexts like classification. —— Taku (talk) 11:36, 22 April 2019 (UTC)

By the way, the last two links are either broken or I cannot find “space curve”. Also, Vakil is using the term “elliptic space curve” (not an isolated instance of “space curve”); if I understand, the intention there seems to emphasize he wants to consider elliptic space curve not elliptic plane curve. —— Taku (talk) 11:42, 22 April 2019 (UTC)

Right, but "projective curve" doesn't describe the same concept as a projective curve can be a plane curve or a space curve. I don't have much preference for the term "space curve" specifically, but I do think that the plane and non-planar curves should be more clearly delineated in algebraic curve with more attention to the non-plane curves than there is now. As to the previous links, they still seem to work for me but here is the arXiv link for the RIMS paper (about space curves) and the JSTOR link & McMullen's website link for the Annals paper (Section 4 is about genus 4 space curves). — MarkH21 (talk) 22:05, 22 April 2019 (UTC)

(Thank you for the links.) I'm with D.Lazard that there is no bright line between a plane curve and a space curve. See the new intro of Draft:Complete algebraic curve. By "curve", if you're thinking of a smooth projective curve, then it matters whether a curve can be embedded into P^2 or not (while every smooth curve lie in P^3.) If you're willing to embedded a curve with some singularities, up to a birational morphism, then it can be embedded into P^2. In other words, whether a curve is plane or space isn't a good well-defined question, at least from the theoretical point of view. I suspect this is part of the reason the term "space curve" is not used much in technical discussions today (as opposed to more leisurely account) if not exactly dated. -- Taku (talk) 23:10, 22 April 2019 (UTC)

The RIMS paper is a good example: "space curve" appears in the title but never in the abstract, where a more precise formulation is used (namely, a (projective) smooth connected curve). Also, the paper is a type of stuff not appropriate for algebraic curve. It requires some (substantial?) background on algebraic geometry while the Wikipedia article is more for the readers without backgrounds in algebraic geometry. And that's the case for an article separate from algebraic curve. -- Taku (talk) 23:16, 22 April 2019 (UTC)

Okay, I suppose the best thing to do is to just expand the section on algebraic space curves within the algebraic curve with how they may be considered differently when we don't want to allow for singularities via birational equivalence. But a standalone article may be too much at this point. — MarkH21 (talk) 21:16, 23 April 2019 (UTC)

But Draft:Complete algebraic curve looks already long enough for a separate article and is set to become much longer. Why too much? Just as we have "projective variety" in addition to "algebraic variety", it is perfectly normal to have a separate article covering the important special case (i.e., projective case); my *view* (but not necessarily the others) is that details are easier to digest if they are spread across different articles than in a single one. -- Taku (talk) 04:33, 24 April 2019 (UTC)

Sorry, I meant algebraic space curve (and algebraic plane curve) do not need standalone articles at this point. Projective curve (or complete curve, but I do think projective curve is more common as I mentioned in Draft talk:Complete algebraic curve) should certainly have its own article. — MarkH21 (talk) 06:35, 24 April 2019 (UTC)

No, no, I completely misunderstood. I will also investigate which one is better "complete curve" or "projective curve" (both should be fine though). -- Taku (talk) 23:32, 25 April 2019 (UTC)

I have edited the lead of Algebraic curve for fixing an error in the first sentence, and making it less technical, that is introducing the non-elementary concept of "algebraic variety of dimension one" as late as possible. By the way, I have introduced the different sorts of algebraic curves (affine, projective, plane or not). This seems necessary, because I believe that most readers who come to this article are searching, in fact, for affine plane algebraic curves.

I have also emphasized that the distinction between affine and projective is not important as one passes easily from one case to the other. More generally, as every curve is birationally equivalent to a plane curve, I tend to agree with MarkH21, that, at this point, we do not need separate articles for different sorts of algebraic curves.

... with the exception that we have already Moduli of algebraic curves. This article must be linked in Algebraic curve, and deserves to have a short section there. IMO, Taku's draft must be merged there, or renamed Algebraic curve (scheme theory). D.Lazard (talk) 09:19, 26 April 2019 (UTC)

I think it's a matter of perspective and that's the point I wanted to make above. If you're thinking "birationally", of course, affine vs projective or smooth or singular isn't important. I don't disagree "one passes easily from one case to the other" (as one can complete an affine one to a projective one). It doesn't follows, though, that the distinction is't important. Is the distinction between "affine variety" and "projective variety" not important? I disagree; the former is mostly about local matters while the latter about the global matter; e.g., the question on an embedding or cohomology. Cohomology or intersection theory does change in a meaningful way going from "affine" to "projective". There are some global aspects do not fit naturally into algebraic curve since the article focuses on the plane and affine cases (which itself is not a wrong choice at all.) For example, I don't think some extended discussion of a canonical curve or linear series or the use of Hilbert scheme makes much sense in algebraic curves. (Update: I have started the section "A map from a curve to a projective space" in the draft to illustrate what I have in mind.)

Perhaps, the main issue is whether to assume some background in algebraic geometry or not. The article "algebraic curve" is currently accessible to those without backgrounds in algebraic geometry and I think we all agree that that accessibility should not be compromised. On the other hand, some readers do have necessary backgrounds, who need/want to see discussion of line bundles or divisors; usually the easiest way to serve two segments of the readers is to use separate articles. "Algebraic curve (scheme theory)" is probably a wrong name (since it's not this is not about scheme theory but global aspect); why is "complete curve" or "projective curve" a wrong one? (note "algebraic curve" has more or less nothing about that topic.) -- Taku (talk) 12:10, 26 April 2019 (UTC)

I have rewritten the lead of curve. This is the first step toward the point 3 of above MarkH21's suggestion (and of the first point of mine). D.Lazard (talk) 09:57, 1 May 2019 (UTC)

(I have zero opinion on whether this article needs to be merged with some others or stand alone.) But, in the broad curve article discussion, one question that has not been touched: what about a family of algebraic curves? I don't think moduli of curves is a place for this. A lengthy discussion of it feels out of place for algebraic curve; that leaves us Draft:complete algebraic curve. Maybe proper curve is a better name for "Draft:complete algebraic curve". -- Taku (talk) 23:43, 1 May 2019 (UTC)