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This is a former exam question:

If, in the projective space $\mathbb{P}^3_\mathbb{R}$, we consider three coplanar lines having one single point in common, what will be the position of the three corresponding lines in the dual projective space? What if we consider three lines that have pairwise intersections in three points?

Here is what I know: Points in the projective space correspond to planes in the dual projective space, lines to lines, and planes to points. Using these facts, I would “translate” the first situation as follows: In the dual projective space, the three lines are uniquely determined by two planes each. As the lines are coplanar, one plane can be chosen to be the same for all three lines. Then, for each line, a second plane has to be chosen from the pencil of planes containing this particular line. So, basically, the position of the lines is the same, only the way they are determined is different in the dual projective space.

Am I correct with this? How would you answer the question?

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Correct.

The dual of "three coplanar lines" is "three concurrent lines". The dual of "having one single point in common" is "havin one common plane. So the dual is indeed "three concurrent lines in a plane".

What about the other question? The dual of "three coplanar lines" is "three concurrent lines" again. The dual of "have pairwise intersections in three points" is "determining three planes pairwise". So in this case the dual is "three concurrent lines which are not in a plane".

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  • $\begingroup$ Perfect, thank you very much! :-) $\endgroup$ – bbrot Jan 17 '17 at 13:14

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