9 missing lines on a specific smooth cubic surface

Question

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?

Answer 1

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.