I know graphs of the form $A x^2 + B xy + C y^2 + D x + E y + F = 0$ are conic sections, in that they are all cross sections of a cone. But what would happen if I changed the degree of the polynomial to 3? Would these be cross sections of a new 3D shape or a 4D shape? If it is possible, please provide a picture or some sort of representation of this shape thanks.
EDIT: Just to make it clear when I ask about the shape I mean the conic section, not the graph itself.
I read the question again. I realize now the implicit mistake. Generically, provided $F$ is reasonably differentiable, any equation of the form $F(x,y) = 0$ gives a curve in the plane. If you change the exponent to say $Ax^3+\cdots +C=0$ then you would still have a curve in the plane. I suppose the question you are really asking: is such a curve interesting in some way which is similar to that of conic sections? I personally am aware of no such results, but, then again, I have not studied classical algebraic geometry in any depth. I can tell you this, the study of eliptic curves concerns the locus of points which satisfies a relatively simple cubic equation. The structure present there is in fact deeply interesting, there is a hidden group structure and apparently some way to take encryption beyond the current technique. See this wikipedia article
So, all of that for just one cubic curve. I think the heart of your question is worthwhile and I hope someone eventually counters this one with a general theory of cubics. I would be happy to see it.