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What is already known about rigid line arrangements? By line arrangement, I mean a unions of lines in $\mathbb{P}^2_{\mathbb{C}}$ with fixed incidences. (Written in notation, I mean a collection of lines $\mathcal{L}$, a collection of points $\mathcal{P}$, and a set of incidences $\mathcal{I}\subset \mathcal{L}\times\mathcal{P}$ that determine which points lie on which lines.)

By rigid, I mean that all the unions of lines that satisfy the conditions of the arrangement are in the closure of the same $PGL_3$ orbit.

For example, we can pick four points $p_1,\ldots,p_4\in\mathbb{P}^2$ and draw the six lines between them. This arrangement is rigid because $PGL_3$ is 4-transitive on points.

For a more interesting example, we can let $p_1,\ldots,p_9$ be the nine flexes of a smooth plane cubic and draw the twelve lines that pass through three of the flexes. A proof of rigidity is given in http://alpha.math.uga.edu/~luca/hesse.pdf, but the idea is to first note that the flexes don't move in Hesse's pencil, so the construction is independent of j-invariant. Then, given a configuration of 12 lines with the same incidences as the Hesse configuration, you find a smooth cubic passing through the 9 points of concurrency, and then use all the relations you have from the 12 lines to show that those 9 points must be flex points.

Unfortunately, my knowledge ends here. I don't know any other examples except for trivial ones, and I couldn't find anything with an internet search.

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You have to be a bit more careful with your definition of arrangements and their rigidity. Once, you are, then "Mnev universality" provides an enormous supply of rigid arrangements in $P^2$ (even on scheme-theoretic level). See Theorem 1.3 in my paper with John Millson "On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties".

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  • $\begingroup$ I am very glad I found this beautiful paper! Thanks for linking it. $\endgroup$ Jul 6, 2020 at 1:47
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Perhaps you are familiar with this survey, which seems relevant:

Felsner, Stefan, and Jacob E. Goodman. "Pseudoline Arrangements." Handbook of Discrete and Computational Geometry, JE Goodman, ed., 3rd ed., CRC Press, Boca Raton, FL (2016). Soon (Oct17) to appear as 4th ed: PDF download.


CRC


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    $\begingroup$ I'm not familiar with the survey, but thanks for the reference. Looking at the introduction, it says that they work in the real projective plane, while I need results in the complex projective plane. Do you think results about psuedolines in the real plane can help with (straight, complex) lines in the complex projective plane? $\endgroup$
    – DCT
    Jul 29, 2017 at 1:08
  • $\begingroup$ @DCT: Many differences; not sure myself if I can detail the essential differences. Hopefully others can... $\endgroup$ Jul 29, 2017 at 1:11
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    $\begingroup$ [Also, presumably "psuedoline" is a typo for "pseudoline", not a new kind of mathematical ojbect.] $\endgroup$ Jul 29, 2017 at 1:32

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