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.)


There are non-classical projective planes which are completely "wild", i.e. there is no hope in getting a meaningful structure that provides a coordinatisation, e.g. free projective planes:

start from a configuration of "points" and "lines" which are a partial linear space, i.e. 2 points are on at most 1 line, and 2 lines intersect in at most 1 point. Then try to complete the picture in stages (at odd stages add missing points, at even stages add missing lines). In the limit one gets a projective plane, containing the initial configuration.

Even in the finite case, there is essentially a zoo of weird examples, cf. the wikipedia; you might want to restrict to ones that have at least a big of structure, e.g. these related to near-fieds, semifields and quasifields.

If you restruct to one of the latter then the algebra governing your equations gets strange, e.g. for near-fields mutiplication is not commutative and distributivity only holds on one side, etc...

  • $\begingroup$ @ DimaPasechnik: there are indeed many wild and rigid examples. (That's part of the game.) Still, I want to consider general projective planes, so with no restrictions on automorphism groups or coordinatizing structure. $\endgroup$
    – THC
    Jul 19 '19 at 13:30
  • 1
    $\begingroup$ Well, it's too wild, but perhaps you might like the approach via "Zariski geometries" due to Hrushovski & Zilber? ams.org/journals/jams/1996-9-01/S0894-0347-96-00180-4/… $\endgroup$ Jul 19 '19 at 15:00

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