# Theorem name or source: a (ruled) surface with a certain number of non-parallel lines is a plane

I vaguely remember from my youth a result concerning scrolls or ruled surfaces. Here is what I remember:

A ruled surface containing at least $n$ non-parallel lines is a plane

In my memory $n=27$, but I am not very sure. Does this remind someone of a correctly stated theorem, does it have a name?

• I'm totally interested in seeing a proof of this or something like it! Aug 6, 2016 at 4:02
• Maybe you're thinking of the result (I'm trusting the Wikipedia page here) that the plane is the only ruled surface which contains at least three distinct lines through each of its points. Aug 6, 2016 at 6:24

The generic smooth cubic surface has 27 lines on it (working in $\Bbb CP^3$). However, it is not a ruled surface. On the other hand, a nonsingular quadric (working over $\Bbb R$ we need a saddle surface or a hyperboloid of one sheet) has two infinite families of pairwise skew lines on it. So I'm not quite sure what the result you're remembering might be.
Now, if you talk about smooth (real analytic) flat ruled surfaces in $\Bbb R^3$, they must be either planes, cones, cylinders, or tangent developables. Cylinders are ruled by parallel lines, but cones and tangent developables have infinitely many non-parallel rulings.