Surfaces in $\mathbb P^3$ not containing any line Let $d \geq 4$. I'm interested by know if there is a surface $S$ of degree $d$ in $\mathbb P^3_{\mathbb C}$ such that $S$ does not contains a line. I know 
I have no idea how to do it.

This is the famous Noether-Lefschetz theorem. The answer is that for a "very general" such surface -- meaning away from a countable union of proper closed subsets in the parameter space of all degree-d surfaces -- the only algebraic curves are complete intersections with other surfaces. In particular, there are no lines, no conics, etc. on such a surface. While this is true in general, Mumford in the 60s or maybe 70s gave a challenge to come up with a specific example of even one quartic surface not containing a line, a challenge that was not met until just a few years ago.
