Is a ruled surface of degree>2 always singular?

Question

Let $X=\mathbb{C}\mathbb{P}^3$ and let $V\subset X$ be a closed algebraic sub variety. By V-is ruled, I mean that for every point in $V$ there is a line passing through it which also lies in V. Suppose now that V is ruled and is a surface of degree $\geq$3. Is $V$ necessarily singular?

Answer 1

Yes, even much more strongly: only finitely many lines can lie on a non-singular variety $V\subset \mathbb P^3$ of degree $d\geq 3$.

Optional complement
That number is well known to be 27 for $d=3$.
For $d\geq 3$ the maximum number of lines on a smooth surface $V\subset \mathbb P^3$ of degree $d$ is a function $l(d)$ shown by Severi to satisfy $l(d)\leq(d-2)(11d-6)$.
This bound is far from sharp: already $l(4)=64\lt (4-2)(11\times 4-6)=76$.
Here is a recent reference on those questions.

Finally, it is elementary to show that a generic smooth surface $V\subset \mathbb P^3$ of degree $d\geq 4$ actually contains no line at all: Shafarevich Theorem 10, page 80.