Combinatorial curves in combinatorial projective planes

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.)