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.
 A: 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.
A: I think that
$$f(x,y,z) = x^4-3 x y+x^2 z+z^3+y^2+5$$
provides an example of a surface in ${\mathbb A}^3$ that does not contain a line in in ${\mathbb A}^3$.
Substitute $x=u_0 + t p_0$, $y = v_0 + t q_0$ and $z = w_0 + t r_0$ in $f(x,y,z)$.
Collect coefficients of powers of $t$:
$$C_4 t^4 + C_3 t^3 + C_2 t^2 + C_1 t + C_0$$
where the $C_i \in {\mathbb Q}[u_0,v_0,w_0,p_0,q_0,r_0]$
Now calculate three grobner bases of $\{C_i\}_{i=0..4} \cup \{h\}$ where $h$ is either $p_0 - 1$, $q_0 - 1$ or $r_0 -1$. All three reduce to the unit ideal, proving that $V(f(x,y,z))$ does not contain a line in ${\mathbb A}^3$.
