# 9 missing lines on a specific smooth cubic surface

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?

• 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.
– user64687
Dec 3, 2014 at 12:10
• 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.
– user64687
Dec 3, 2014 at 12:13
• 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?
– user64687
Dec 3, 2014 at 12:14
• 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! Dec 3, 2014 at 23:20