TITLE: Picard groups of quartic K3 surfaces
QUESTION [8 upvotes]: Does anyone know where I can find examples of quartic K3 surfaces for which the Picard group is known? I'm really interested in examples where there are explicit constructions of the divisors generating the Picard group (rather than examples where we only know the Picard rank). I appreciate that this is not easy to compute in general, so any useful references you can suggest will be most welcome.

REPLY [2 votes]: There are some simple such examples if you want to know the generators but not explicit equations for the surfaces:

A very general quartic in $\mathbb P^3$ has Picard number $1$ and hence its Picard group is generated by any minimal degree curve.
A quartic, very general among those containing a fixed line has Picard number $2$ and its Picard group is generated by that line and the hyperplane class (or the complementary cubic curve).
A quartic, very general among those containing a fixed conic has Picard number $2$ and its Picard group is generated by that conic and the hyperplane class (or the complementary conic).
A quartic, very general among those containing two fixed skew lines has Picard number $3$ and its Picard group is generated by those lines and the hyperplane class.

If you are only interested in generating $\mathrm{Pic}\otimes\mathbb Q$, then you have that

$\mathrm{Pic}\otimes\mathbb Q$ is generated by the smooth rational and elliptic curves on it for any $K3$ with Picard number at least $5$.
$\mathrm{Pic}\otimes\mathbb Q$ is generated by the smooth rational curves on it for any $K3$ with Picard number at least $12$.