TITLE: no lines/conics on a degree 4/5 surface?
QUESTION [8 upvotes]: Are there explicit examples of smooth surfaces of degree 4 and 5 in P^3, which 
contain no lines or conics (i.e. smooth rational curves of degrees 1 and 2)? An example over a finite field would be better, but if not - I will take one over complex numbers :)

REPLY [3 votes]: A quartic surface containing a line or a conic has Picard number at least 2. There are explicit examples of Ronald van Luijk of quartic surfaces defined over $\mathbb{Q}$ such that the (geometric) Picard number is 1. (See http://www.math.leidenuniv.nl/~rvl/ps/picone.pdf)
The reductions of these surfaces mod 2 and 3 contain a line resp, a conic, so they do not work for you. 
However, this paper hints at algorithm how to find examples in positive characteristic:
Take a smooth quartic surface $S$ over a finite field with $q$ elements. By point counting (or more efficient methods) you can determine the characteristic polynomial $P_2(T)$ of Frobenius on the second (etale) cohomology of $S$. 
Let $r$ be the number reciprocal zeros of $P_2$ the form $q\zeta$ with $\zeta$ a root of unity. This number $r$ has the same parity as the degree of $P_2$, which is $22$ for quartic surfaces. It is well-known that $r$ is at most the geometric Picard number of $S$.
Suppose now that you find a surface such that $r=2$. Then either the geometric Picard number $\rho(S)$ equals 1, and you are done, or the geometric Picard number equals 2. In the latter case the Tate-conjecture holds. Milne proved that the Tate-conjecture implies the Artin-Tate conjecture. Now the Artin-Tate conjecture enables you to determine the discriminant $D$ of the intersection pairing on the Neron-Severi group up to squares from $P_2(T)$. (See, e.g. my paper on elliptic K3 surfaces with geometric Mordell-Weil rank 15.) Now if $\rho(S)=2$ and $S$ contains a line then $D=-k^2$ for some integer $k$, and if $S$ contains a conic then $D=-3k^2$ for some integer $k$ (this is an easy exercise). Hence you have a criterion to check whether your quartic surface contains a line or a conic. 
For quintics something similar works, only the degree of $P_2$ differs, and the values of $D$ in the case that $S$ contains a line or a quintic differ.