Suppose $S\subseteq \mathbb{P}^3(x:y:z:w)$ is a surface of degree $d$, which I assume to be integral as a scheme. So $S$ is the zero locus of a homogeneous polynomial in $k[x,y,z,w]$ of degree $d$ which generates a prime ideal. Here $k$ is the base field and is assumed to be algebraically closed.
If $S$ is ruled (every point of $S$ is contained in a line $L\subseteq \mathbb{P}^3$ which is contained in $S$) and $d>2$, must $S$ be a cone? If not, is there a classification of such surfaces?
The only smooth ruled surfaces are planes and smooth quadrics. So any surface satisfying the above conditions must be singular, but beyond that I've been having trouble determining further properties they must satisfy.
Here's one for you to play with: Consider the surface
$$z^2w^2+4x^3z-6xyzw+4y^3w - 3x^2y^2 = 0.$$
It's called a tangent developable, the surface of lines tangent to a smooth curve in $\Bbb P^3$. Can you find the curve?