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2 broken links fixed, cf. https://math.meta.stackexchange.com/a/34713/228959
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Glorfindel
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I believe the answer is (in general) No for the following reason. A configuration of points in $\mathbb{P}^2$ is projectively dual to an arrangement of lines in $\mathbb{P}^2$. The question you ask, translated to arrangements of lines, is whether an arrangement can always be continuously moved to any isomorphic arrangement, all the while remaining isomorphic. This was asked by Ringle in 1956 (the isotopyisotopy conjecture), and answered negatively by Mnev's Universality TheoremMnev's Universality Theorem in 1985: not only can the space of line arrangments isomorphic to a given arrangement be disconnected, it can have the homotopy type of any algebraic variety.

Of course it is not difficult to change from $\mathbb{P}^2$ to $\mathbb{R}^2$. I find this explicit claim in a 1988 paper by Suvorov (Springer linkSpringer link):
      Suvorov
Here, "nondegenerated" means, I believe, what we would call today "simple": no three lines share a point. A "rigid" isotopy keeps the lines straight (in a "flexible" isotopy, the objects are pseuodolines).

Suvorov's example has 14 lines in $\mathbb{P}^2$. Later (1996) Richter-Gebert found another 14-line/point example, presented in his paper "Two Interesting Oriented Matroids" (CiteSeer linkCiteSeer link):
      Jurgen R-G Fig 1
He shows that

[its] realization space [...] is an open interval from which one point has been deleted.

I believe the answer is (in general) No for the following reason. A configuration of points in $\mathbb{P}^2$ is projectively dual to an arrangement of lines in $\mathbb{P}^2$. The question you ask, translated to arrangements of lines, is whether an arrangement can always be continuously moved to any isomorphic arrangement, all the while remaining isomorphic. This was asked by Ringle in 1956 (the isotopy conjecture), and answered negatively by Mnev's Universality Theorem in 1985: not only can the space of line arrangments isomorphic to a given arrangement be disconnected, it can have the homotopy type of any algebraic variety.

Of course it is not difficult to change from $\mathbb{P}^2$ to $\mathbb{R}^2$. I find this explicit claim in a 1988 paper by Suvorov (Springer link):
      Suvorov
Here, "nondegenerated" means, I believe, what we would call today "simple": no three lines share a point. A "rigid" isotopy keeps the lines straight (in a "flexible" isotopy, the objects are pseuodolines).

Suvorov's example has 14 lines in $\mathbb{P}^2$. Later (1996) Richter-Gebert found another 14-line/point example, presented in his paper "Two Interesting Oriented Matroids" (CiteSeer link):
      Jurgen R-G Fig 1
He shows that

[its] realization space [...] is an open interval from which one point has been deleted.

I believe the answer is (in general) No for the following reason. A configuration of points in $\mathbb{P}^2$ is projectively dual to an arrangement of lines in $\mathbb{P}^2$. The question you ask, translated to arrangements of lines, is whether an arrangement can always be continuously moved to any isomorphic arrangement, all the while remaining isomorphic. This was asked by Ringle in 1956 (the isotopy conjecture), and answered negatively by Mnev's Universality Theorem in 1985: not only can the space of line arrangments isomorphic to a given arrangement be disconnected, it can have the homotopy type of any algebraic variety.

Of course it is not difficult to change from $\mathbb{P}^2$ to $\mathbb{R}^2$. I find this explicit claim in a 1988 paper by Suvorov (Springer link):
      Suvorov
Here, "nondegenerated" means, I believe, what we would call today "simple": no three lines share a point. A "rigid" isotopy keeps the lines straight (in a "flexible" isotopy, the objects are pseuodolines).

Suvorov's example has 14 lines in $\mathbb{P}^2$. Later (1996) Richter-Gebert found another 14-line/point example, presented in his paper "Two Interesting Oriented Matroids" (CiteSeer link):
      Jurgen R-G Fig 1
He shows that

[its] realization space [...] is an open interval from which one point has been deleted.

Image links broken; now fixed.
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Joseph O'Rourke
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I believe the answer is (in general) No for the following reason. A configuration of points in $\mathbb{P}^2$ is projectively dual to an arrangement of lines in $\mathbb{P}^2$. The question you ask, translated to arrangements of lines, is whether an arrangement can always be continuously moved to any isomorphic arrangement, all the while remaining isomorphic. This was asked by Ringle in 1956 (the isotopy conjecture), and answered negatively by Mnev's Universality Theorem in 1985: not only can the space of line arrangments isomorphic to a given arrangement be disconnected, it can have the homotopy type of any algebraic variety.

Of course it is not difficult to change from $\mathbb{P}^2$ to $\mathbb{R}^2$. I find this explicit claim in a 1988 paper by Suvorov (Springer link):
      Suvorov http://cs.smith.edu/%7Eorourke/MathOverflow/Suvorov13.jpgSuvorov
Here, "nondegenerated" means, I believe, what we would call today "simple": no three lines share a point. A "rigid" isotopy keeps the lines straight (in a "flexible" isotopy, the objects are pseuodolines).

Suvorov's example has 14 lines in $\mathbb{P}^2$. Later (1996) Richter-Gebert found another 14-line/point example, presented in his paper "Two Interesting Oriented Matroids" (CiteSeer link):
      Jurgen R-G Fig 1 http://cs.smith.edu/%7Eorourke/MathOverflow/JurgenR-G14.jpgJurgen R-G Fig 1
He shows that

[its] realization space [...] is an open interval from which one point has been deleted.

I believe the answer is (in general) No for the following reason. A configuration of points in $\mathbb{P}^2$ is projectively dual to an arrangement of lines in $\mathbb{P}^2$. The question you ask, translated to arrangements of lines, is whether an arrangement can always be continuously moved to any isomorphic arrangement, all the while remaining isomorphic. This was asked by Ringle in 1956 (the isotopy conjecture), and answered negatively by Mnev's Universality Theorem in 1985: not only can the space of line arrangments isomorphic to a given arrangement be disconnected, it can have the homotopy type of any algebraic variety.

Of course it is not difficult to change from $\mathbb{P}^2$ to $\mathbb{R}^2$. I find this explicit claim in a 1988 paper by Suvorov (Springer link):
      Suvorov http://cs.smith.edu/%7Eorourke/MathOverflow/Suvorov13.jpg
Here, "nondegenerated" means, I believe, what we would call today "simple": no three lines share a point. A "rigid" isotopy keeps the lines straight (in a "flexible" isotopy, the objects are pseuodolines).

Suvorov's example has 14 lines in $\mathbb{P}^2$. Later (1996) Richter-Gebert found another 14-line/point example, presented in his paper "Two Interesting Oriented Matroids" (CiteSeer link):
      Jurgen R-G Fig 1 http://cs.smith.edu/%7Eorourke/MathOverflow/JurgenR-G14.jpg
He shows that

[its] realization space [...] is an open interval from which one point has been deleted.

I believe the answer is (in general) No for the following reason. A configuration of points in $\mathbb{P}^2$ is projectively dual to an arrangement of lines in $\mathbb{P}^2$. The question you ask, translated to arrangements of lines, is whether an arrangement can always be continuously moved to any isomorphic arrangement, all the while remaining isomorphic. This was asked by Ringle in 1956 (the isotopy conjecture), and answered negatively by Mnev's Universality Theorem in 1985: not only can the space of line arrangments isomorphic to a given arrangement be disconnected, it can have the homotopy type of any algebraic variety.

Of course it is not difficult to change from $\mathbb{P}^2$ to $\mathbb{R}^2$. I find this explicit claim in a 1988 paper by Suvorov (Springer link):
      Suvorov
Here, "nondegenerated" means, I believe, what we would call today "simple": no three lines share a point. A "rigid" isotopy keeps the lines straight (in a "flexible" isotopy, the objects are pseuodolines).

Suvorov's example has 14 lines in $\mathbb{P}^2$. Later (1996) Richter-Gebert found another 14-line/point example, presented in his paper "Two Interesting Oriented Matroids" (CiteSeer link):
      Jurgen R-G Fig 1
He shows that

[its] realization space [...] is an open interval from which one point has been deleted.

added 98 characters in body
Source Link
Joseph O'Rourke
  • 150.8k
  • 36
  • 358
  • 958

I believe the answer is (in general) No for the following reason. A configuration of points in $\mathbb{P}^2$ is projectively dual to an arrangement of lines in $\mathbb{P}^2$. The question you ask, translated to arrangements of lines, is whether an arrangement can always be continuously moved to any isomorphic arrangement, all the while remaining isomorphic. This was asked by Ringle in 1956 (the isotopy conjecture), and answered negatively by Mnev's Universality Theorem in 1985: not only can the space of line arrangments isomorphic to a given arrangement be disconnected, it can have the homotopy type of any algebraic variety.

Of course it is not difficult to change from $\mathbb{P}^2$ to $\mathbb{R}^2$. I find this explicit claim in a 1988 paper by Suvorov (Springer link):
      Suvorov http://cs.smith.edu/%7Eorourke/MathOverflow/Suvorov13.jpg
Here, "nondegenerated" means, I believe, what we would call today "simple": no three lines share a point. A "rigid" isotopy keeps the lines straight (in a "flexible" isotopy, the objects are pseuodolines).

Suvorov's example has 14 lines in $\mathbb{P}^2$. Later (1996) Richter-Gebert found another 14-line/point example, presented in his paper "Two Interesting Oriented Matroids" (CiteSeer link):
      Jurgen R-G Fig 1 http://cs.smith.edu/%7Eorourke/MathOverflow/JurgenR-G14.jpg
He shows that

[its] realization space [...] is an open interval from which one point has been deleted.

I believe the answer is (in general) No for the following reason. A configuration of points in $\mathbb{P}^2$ is projectively dual to an arrangement of lines in $\mathbb{P}^2$. The question you ask, translated to arrangements of lines, is whether an arrangement can always be continuously moved to any isomorphic arrangement, all the while remaining isomorphic. This was asked by Ringle in 1956, and answered by Mnev's Universality Theorem in 1985: not only can the space of line arrangments isomorphic to a given arrangement be disconnected, it can have the homotopy type of any algebraic variety.

Of course it is not difficult to change from $\mathbb{P}^2$ to $\mathbb{R}^2$. I find this explicit claim in a 1988 paper by Suvorov (Springer link):
      Suvorov http://cs.smith.edu/%7Eorourke/MathOverflow/Suvorov13.jpg
Here, "nondegenerated" means, I believe, what we would call today "simple": no three lines share a point. A "rigid" isotopy keeps the lines straight (in a "flexible" isotopy, the objects are pseuodolines).

Suvorov's example has 14 lines in $\mathbb{P}^2$. Later (1996) Richter-Gebert found another 14-line/point example, presented in his paper "Two Interesting Oriented Matroids" (CiteSeer link):
      Jurgen R-G Fig 1 http://cs.smith.edu/%7Eorourke/MathOverflow/JurgenR-G14.jpg
He shows that

[its] realization space [...] is an open interval from which one point has been deleted.

I believe the answer is (in general) No for the following reason. A configuration of points in $\mathbb{P}^2$ is projectively dual to an arrangement of lines in $\mathbb{P}^2$. The question you ask, translated to arrangements of lines, is whether an arrangement can always be continuously moved to any isomorphic arrangement, all the while remaining isomorphic. This was asked by Ringle in 1956 (the isotopy conjecture), and answered negatively by Mnev's Universality Theorem in 1985: not only can the space of line arrangments isomorphic to a given arrangement be disconnected, it can have the homotopy type of any algebraic variety.

Of course it is not difficult to change from $\mathbb{P}^2$ to $\mathbb{R}^2$. I find this explicit claim in a 1988 paper by Suvorov (Springer link):
      Suvorov http://cs.smith.edu/%7Eorourke/MathOverflow/Suvorov13.jpg
Here, "nondegenerated" means, I believe, what we would call today "simple": no three lines share a point. A "rigid" isotopy keeps the lines straight (in a "flexible" isotopy, the objects are pseuodolines).

Suvorov's example has 14 lines in $\mathbb{P}^2$. Later (1996) Richter-Gebert found another 14-line/point example, presented in his paper "Two Interesting Oriented Matroids" (CiteSeer link):
      Jurgen R-G Fig 1 http://cs.smith.edu/%7Eorourke/MathOverflow/JurgenR-G14.jpg
He shows that

[its] realization space [...] is an open interval from which one point has been deleted.

added 72 characters in body
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Joseph O'Rourke
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  • 358
  • 958
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Joseph O'Rourke
  • 150.8k
  • 36
  • 358
  • 958
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Joseph O'Rourke
  • 150.8k
  • 36
  • 358
  • 958
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