Let $C$ be an $(n_r, b_k)$ combinatorial configuration that admits a geometric realization in the plane. I'm interested in the maximum number of points/lines $M$ of $C$ we can place in general position on the plane, before the remainder of the configuration is completely determined. Alternatively, given a geometric embedding, how many degrees of freedom does it have (where only the incidence structure must be maintained) to 'wiggle' around locally. I understand these are different questions.
I imagine this question would be difficult for an arbitrary configuration. I'm more looking for an upper bound on $M$ that depends on $n,r,b,k$, and perhaps other purely combinatorial parameters.
I've looked at many of the commonly cited sources on the subject of geometric configurations, but I haven't seen anything that relates to this question.
I get a sense that the connectivity of the Levi graph might play some sort of role in bounding this number. I've built simple examples (analyzing the Levi graphs of things like a configuration $C$ with an extra point joined to every vertex, or taking the $k$-fold product of a configuration), but any trends I see are difficult to pin down in general.
Another rather 'trivial' thing I've noticed is that if the configuration is embedded, and we try to move a point $x$ of the configuration a small amount in any direction, then at least $r-1$ lines through $x$ must also move. i.e. if a point is moved, then $r-1$ lines through that point move, and similarly if a line moves from its original position, then $k-1$ points through that line move as well. I've used this idea in the following fashion: suppose we fix three points in an embedded configuration $C$, and let $x$ be another point that is able to be locally moved without destroying the configuration. Then the points/lines that move when $x$ moves form a configuration with replication numbers $r-1,r$ and block sizes $k-1,k$. But the problem from here seems to be that two points moved independently seems to provide nothing stronger (it doesn't look like we can 'predict' what happens in any sense).
I know my exterminations are nothing concrete, and a lot of it is just play, but I'd appreciate any ideas on where to look.
Thanks!
G. Flowers Edit: @Paseman: I have indeed thought about configurations in multiple dimensions projecting onto $\mathbb{R}^2$. However, I don't see much of a relation between a planar realization allowing for $M$ points to be placed in general position, and a configuration that is realizable in $\mathbb{R}^d$ (i.e. a $d-$dimensional configuration). Yes, there is some freedom in how we project, but this seems to provide a lower bound on $M$, rather than an upper one.
I apologize for not being immensely clear. I'm referring to point-line configurations in the Euclidean plane; although, the real projective plane works just as well.
As an example, I would say that $M=5$ for the Pappus Configuration, since five points can be placed in general position, and we can still draw the Pappus configuration with them.
