7
$\begingroup$

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.)

$\endgroup$
1
$\begingroup$

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...

$\endgroup$
2
  • $\begingroup$ @ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.