Let $\Gamma (x,y,z) = 27 x^3 + 243 x^2 y+324 x y^2 + 189 y^3 +27 x^2 z + 27 x y z - 27 y^2 z + z^3$. $S: \Gamma (x,y,z) = 27 $ is a smooth cubic surface. Consider lines of the form $x = x_0 + p s$, $y = y_0 + s$, $z = z_0 + q s$ on the surface. The Caley-Salmon theorem says there are 27 such lines defined over the complex numbers but I can only find $18$. I get these $18$ lines by substitution of the $x$, $y$, $z$ of the line into $\Gamma (x,y,z) = 27 $ since the coefficients of $s$ should vanish. The resulting equations are \begin{eqnarray} \Gamma (p, 1, q) & = & 0 , \\ \Gamma_x (p, 1, q) x_0 + \Gamma_y (p, 1, q) y_0 + \Gamma_z (p, 1, q) z_0 & = & 0 , \\ \Gamma_x (x_0, y_0, z_0) p + \Gamma_y (x_0, y_0, z_0) + \Gamma_z (x_0, y_0, z_0) q & = & 0 , \\ \Gamma (x_0, y_0, z_0) & = & 27 . \end{eqnarray} Taking the resultant of the left hand sides of the first two equations gives $3$ possible values for the pairs $(p, q) \in \mathbb{C}^2 $. Each line should pass through the plane $z = 0$ and so each pair $(p, q)$ gives $6$ of the points $(x_0, y_0, 0)$ the line should pass through since the third equation is quadratic and the fourth is cubic. I count only $18 $ lines. What have I missed?

  • $\begingroup$ Something about your definining equation doesn't make sense. It is a homogeneous cubic in 3 variables. So it either corresponds to a smooth projective cubic curve $C \subset \mathbf P^2$, or else to an affine cubic surface $\tilde{C} \subset \mathbf A^3$, the cone over $C$. But in the latter case the surface is definitely not smooth --- like any cone, it has a singular point at the origin. $\endgroup$ – user64687 Dec 3 '14 at 12:10
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    $\begingroup$ Oops, I missed the "=27" part, so what I said about being a cone is rubbish. Nevertheless, the fact that this is an affine surface means that Cayley--Salmon cannot give you what you want directly: there may be missing lines at infinity. $\endgroup$ – user64687 Dec 3 '14 at 12:13
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    $\begingroup$ Another possible source of missing lines: you don't seem to allow your lines to have constant $y$-coordinate. How do you know there are no such lines? $\endgroup$ – user64687 Dec 3 '14 at 12:14
  • $\begingroup$ Thanks Asal. I've checked y = a constant and found another 18 lines. It seems that some must be the same line but it's not so easy to recognise when they are all complex! $\endgroup$ – Samuel Hambleton Dec 3 '14 at 23:20

Your 3 missing lines are at infinity and your cubic surface is singular so the Cayley-Salmon theorem does not apply. You have 3 lines in total.

  • $\begingroup$ When you say "your" who are you referring to? Yourself? ;) $\endgroup$ – Cheerful Parsnip Dec 14 '14 at 23:32
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    $\begingroup$ Yes :) This surface has driven me to talk to myself. $\endgroup$ – Samuel Hambleton Dec 15 '14 at 6:28

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