TITLE: Lines on a singular cubic surface
QUESTION [6 upvotes]: How many lines the cubic surfaces $xyz=w^3 \in \mathbb P^3$ has? I found only three: $x=w=0$, $y=w=0$ and $z=w=0$. How to prove that there are no other lines? Also, this surface is singular, is it possible to calculate these lines with some natural multiplicities, summing up to 27?

REPLY [2 votes]: A line not in the plane at infinity has a parametric equation of the form
$$x= x_0 + a t\\
y = y_0+ b t\\
z = z_0 + c t$$
If it were contained in the (finite) part of this surface $x y z = 1$, then we would have
$$(x_0 + a t)(y_0 + b t)(z_0 + c t) = 1$$
for every $t$. Note that $a$, $b$,$c$ are not all $0$, so on the LHS we have a polynomial in $t$ of degree $\ge 1$. We get a contradiction.