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If we consider 3 skew lines and the hyperboloid formed by them, any other line on the hyperboloid intersects these three lines. Moreover, the lines orthogonal to the plane containing a point on the hyperboloid and a line among the 3 skew lines belong to a linear line complex.

In case of a cubic surface, there are two reguli of 3 skew lines on them. Apart from that, are there any other special relations between the 27 lines? I was wondering about this question because I am trying to retrieve a cubic surface by just knowing three skew lines (well I know three more because of the complementary reguli) on it. Is it possible?

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    $\begingroup$ mathoverflow.net/q/257313/30186 $\endgroup$
    – Wojowu
    Commented Oct 30, 2020 at 15:22
  • $\begingroup$ I just address the question about three skew lines (and ignore the parenthetical because I don't know what "complementary reguli" means). A very simple parameter shows the answer is negative. Vanishing on a line imposes 4 conditions on a cubic form. So vanishing on 3 skew lines imposes (at most) 12 conditions. But the space of cubic forms has dimension 20. $\endgroup$
    – Pop
    Commented Oct 30, 2020 at 15:38
  • $\begingroup$ @Wojowu Thank you! That's a useful link. $\endgroup$
    – AUNebulosa
    Commented Oct 30, 2020 at 17:13
  • $\begingroup$ @Pop By complementary reguli, I mean the set of three other skew lines which also lie on the cubic surface and pairwise intersect the original three skew lines. So, if I have 6 lines, each pair of intersecting lines must yield 7 independent conditions on the cubic form, thus three pairs impose 21 conditions (at most). Shouldn't I be able to determine the cubic this way? $\endgroup$
    – AUNebulosa
    Commented Oct 30, 2020 at 17:21

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