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I often heard that each smooth cubic surface contains even $27$ straight lines, such as this statement. So I use the Mathematica plot the equation with this code:$$\begin{align}1 + 54 x y z - &9 (x + y + z) + 126 (x y + x z + y z) - 9 (x^2 + y^2 + z^2) - \\&189 (x^2 y + x y^2 + x^2 z + y^2 z + x z^2 + y z^2) + 81 (x^3 + y^3 + z^3)=0\end{align}$$

But when I solve a cubic equation, like $x^3 + 3 y^3 + z^3-2 x^2 + 5 x y - x + 7=0$ Now I only just find $3$ lines:

enter image description here

Is this surface not smooth? Or is the theorem wrong? Or am I getting it wrong? How to understand this phenomenon?

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  • $\begingroup$ The 27 lines are guaranteed in complex space, looks like. They will not all necessarily be real, and thus visible. $\endgroup$ Feb 3, 2023 at 8:40
  • $\begingroup$ @DanUznanski I don't think I've seen that request(complex space) anywhere else. All I saw was a book that said "smooth" $\endgroup$
    – mayi
    Feb 3, 2023 at 9:23
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    $\begingroup$ Wikipedia uses the phrase "algebraically closed field" which describes complex numbers but not real numbers. $\endgroup$ Feb 3, 2023 at 9:30
  • $\begingroup$ @DanUznanski What condition does the cubic equation satisfy when it has 27 real lines in $\endgroup$
    – mayi
    Feb 3, 2023 at 10:02
  • $\begingroup$ Look at blogs.ams.org/visualinsight/2016/02/15/…. $\endgroup$ Feb 3, 2023 at 10:13

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