Suppose $\mathcal{P}$ is an axiomatic projective plane (a nondegenerate point-line incidence geometry in which there is a unique line on any two different points, and a unique intersection point for any two different lines).

Such a plane is an abstraction of planes $\mathbb{P}^2(k)$ with $k$ a field. Or of $\mathrm{Proj}(k[x,y,z])$ if you want.

Now in such classical planes, we can define *projective curves* by equations.

My question is: what is the best known approach to *combinatorial projective curves* in axiomatic planes $\mathcal{P}$?

If $\mathcal{C}$ is a projective curve in $\mathbb{P}^2(k)$ of degree $m$, then its $k$-rational points have the property that each $k$-line of the plane contains at most $m$ such points, so we could make abstraction of this combinatorial property. But that is too general, and I want to look (far) beyond this step.

(I only know of Manin's fundamental work on combinatorial *cubic* curves (and surfaces), and some work of Buekenhout.)