# How many cubic curves are there?

It is well-known that there is only one "kind" of line, and that there are three "kinds" of quadratic curves (the nature of which depends on the sign of a so-called "discriminant").

It is noteworthy that many of the named cubic curves look rather similar: the folium of Descartes, the trisectrix of Maclaurin, the (right) strophoid, and the Tschirnhausen cubic look very similar in form; the semicubical parabola and the cissoid of Diocles resemble each other as well.

I have deliberately placed the word "kind" in quotes since there does not seem to me an intuitive way of defining the term, so an answer to my question might have to define "kind" rigorously in the context of cubic curves. (An algebraic invariant, for instance... it is a pity that there does not seem to be an analogue of "eccentricity" for cubics!)

In here, it is noted that Newton classified cubics into 72 "kinds", and Plücker after him described 219 "kinds".

So, how does one algebraically distinguish one cubic curve from another, and with a rigorous definition of "kind", how many cubics are there?

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It all depends on what you want to do. Do you want to talk about, say, isomorphism as real affine varieties? Or as (possibly singular) manifolds? Etc. etc. –  Qiaochu Yuan Dec 20 '10 at 3:30
@Qiaochu: Ah, I knew there was a caveat like that... could you start with those two? For everybody: some more detail about the history of Newton and Plücker would also be appreciated. –  Ｊ. Ｍ. Dec 20 '10 at 3:34
These are all curves of the form y^2=P(x)/Q(x), where P is cubic and Q is either linear or constant. Is this the universe you want to consider? –  Ross Millikan Dec 20 '10 at 3:42
@Ross: I'm not (consciously at least) limiting my considerations; for instance, how would $x^3+xy^2+x^2y+y^3=c$ be "classified"? –  Ｊ. Ｍ. Dec 20 '10 at 3:55
I didn't see any examples in your list that had y^3. Maybe I missed them. Maybe that restricts what they look like. –  Ross Millikan Dec 20 '10 at 3:59
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The most standard notion of "kind" is that of isomorphism in algebraic geometry, that is, there are polynomial maps from one curve to the other which, when composed, are the identity on the curves (they don't have to be the identity in other places).

For real, affine plane curves, this notion reproduces the fact that there is a unique line and exactly three smooth conics (over the complex numbers, ellipses and hyperbolas turn out to be the same, and in the projective plane, all are the same).

However, regardless of ground field (well, regardless of working over the real or complex numbers), there are infinitely many different cubics up to isomorphism, determined by the j-invariant, one for every element of the field.

A conclusion you can draw from this is that there really isn't a good algebraic way to say that there are only finitely many types of plane cubic, unless you want to lump all smooth cubics together (which you clearly don't, as you consider parabolas and hyperbolas and ellipses to be distinct). I don't know what method Newton and Plücker used to classify them, but isomorphism is the standard method in modern algebraic geometry.

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As you can probably tell, algebraic geometry isn't my expertise... :( so you're effectively saying there's no "nice" algebraic invariant with a geometric interpretation (analogous to eccentricity for conics) that can be used to distinguish one cubic from another? –  Ｊ. Ｍ. Dec 20 '10 at 3:53
I'm saying that I'm not aware of one that behaves like eccentricity...and that eccentricity is, in some sense, not an algebraic invariant, in that algebraically we can't really distinguish the various conics from each other in any decent way. There may be some other geometric invariant that will behave that way, but it won't, in some sense, be algebraic. –  Charles Siegel Dec 20 '10 at 4:00