which algebraic curves admit (nice?) geometric characterizations? Caveat lector: this a super soft question.
In an effort to help my students, across many different courses, better appreciate and put to work the distinction between geometric objects & properties that are "intrinsic" or "coordinate-independent" and algebraic objects & properties that depend on "extrinsic" definition by coordinates, I thought I would compile a gallery of examples of algebraic curves commonly encountered in the high school curriculum (graphs of low-degree polynomials, conic sections, etc.) -- something along the lines of "Four Ways of Describing Curves in the Plane," emphasizing that a single geometric object admits many different algebraic descriptions (viz.: as an image of a function in only the most special cases, an image of a parametrization more broadly, or as a level set even more broadly) that depend on coordinates. I think this sort of thing should become routine in high school courses, not least because my students in linear algebra never fail to struggle with the idea of a linear transformation as an object that exists independent of its representation with respect to coordinates in a specific basis. That struggle really isn't their fault. They've been taught to over-algebraize everything and conceive of the objects they study as essentially rather than accidentally attached to a specific coordinate description.
So I thought I'd write up a workbook that shows students how to take several of the geometric objects they encounter routinely and (a) describe them algebraically in as many ways as possible in standard Cartesian coordinates, (b) describe them algebraically in as many ways as possible in OTHER coordinates (with emphasis on simple translated coordinates for their importance in completing the square and the Tschirnhaus transformation for eliminating quadratic terms in cubics, etc.), and (c) describe them geometrically, independent of coordinates. So you can imagine a nice little exercise here, taking the parabola given by $f(x)=x^2+2x+1$, say, and forcing to students to work AWAY from this overly familiar representation through a parametric and level-set description, to a description in translated coordinates, and finally to the holy grail, the geometric description in terms of focus and directrix. I think motivating the discussion as one that aims to try to move AWAY from algebra and coordinates TOWARD a characterization that uses purely geometrical ideas would help students understand why they have to learn all that crap about directrices anyway: they could at least learn to see what makes it conceptually interesting, instead of just seeing it as yet another way of describing an object they already know how to describe. What isn't emphasized enough, of course, is the nature of the description. The goal would be to help students start to conceive of the graph of a function in the plane as an equivalence class of many different functions, namely those that can be obtained from the original function by a change of coordinates. And you can see how nicely this would help students start thinking in terms of Lie group actions, etc....
But now, here's the thing. I've never had any algebraic geometry. I wanted to take some cubic polynomials through this series of redescriptions, especially for the sake of showing students that a translation can always eliminate the quadratic and hence of illustrating that an intelligent change of coordinates can greatly simplify a problem. But then I realized: I don't know a geometric characterization of graphs of general cubics. And then I realized it's even worse than that: I don't even know in general which algebraic curves admit geometric characterizations (where a geometric characterization would rely solely on metric notions without giving specific coordinates). In fact, I know that's an ill-posed problem, but I imagine the basic idea is a driving idea behind a lot of work in algebraic geometry.
So my questions are pretty simple and perhaps trivial for people who know algebraic geometry, so I apologize in advance, but I hope I might be able to elicit some interesting answers/comments:


*

*(1) is there some nice way to characterize graphs of cubics geometrically?

*(2) what sense can be made of my question about "which algebraic curves admit geometric characterizations?" My gut says something like -- every curve should admit one; it just might be incredibly hard/impossible to describe it.

 A: First question

(1) is there some nice way to characterize graphs of cubics geometrically?

By “graph of cubic” I assume you mean something like $y = ax^3 + bx^2 + cx + d$, right? In this form the $x$ and $y$ directions play different roles, which makes them a rather coordinate-dependent thing. The proper geometric generalization would probably be
$$
a_{3,0}x^3 +
a_{2,0}x^2 +
a_{2,1}x^2y +
a_{1,0}x +
a_{1,1}xy +
a_{1,2}xy^2 +
a_{0,0} +
a_{0,1}y +
a_{0,2}y^2 +
a_{0,3}y^3
= 0$$
You can recover your version by setting all coefficients related to $y$ to zero, except for $a_{0,1}$ which you'd set to $-1$. So this is more general.
There might be better references to this, but Cayley's paper on the construction of the ninth point of intersection of the cubics which pass through eight given points does start by describing a construction for a cubic given 9 points which define it. It then continues to the special case where this cubic is still not uniquely defined, because the 9th point is already implied by the other 8 due to Cayley-Bacharach theorem. All of this pretty independent of coordinate system. It might contain useful bits and pointers.
Second question

(2) what sense can be made of my question about "which algebraic curves admit geometric characterizations?" My gut says something like -- every curve should admit one; it just might be incredibly hard/impossible to describe it.

By fixing a projective reference frame, i.e. a line at infinity, an origin and a point a unit distance away from the origin in both coordinate directions, you can turn any algebraic computation on the coordinates of a point into a sequence of constructions. This is because there exist constructions to perform addition and multiplication using primitives from projective incidence geometry. So a formulation like “a point lies on this curve if that equation is zero” can be turned into “a point lies on this curve if that point constructed from it coincides with the origin”. If the original curve is independent of coordinate system, then the reference frame can be chosen as arbitrarily. In particular, if a class of curves is invariant under projective transformations, then the projective reference frame may be chosen completely arbitrarily. If, on the other hand, your curves are only invariant under e.g. affine transformations, then you have to fix the role of the line at infinity, which means adding things like parallel lines to your set of geometric primitives. Likewise for similarity transformations, where you need primitives related to angles, or even Euclidean transformations, where you need length measurements.
