Let $V \rightarrow \mathbb P^2$ be the blow-up at two distinct points. ($V$ is a Del Pezzo surface of degree 7.) Choose a smooth curve $C$ from the linear system $|-2K_V|$ and let $S \rightarrow V$ be the double cover, branched along $C$. Then $S$ is a K3 surface whose Picard rank is at least three.

Assume that $S$ is of Picard rank $three$.

My question is:

Can $S$ be embedded into $\mathbb P^3$?

This question is equivalent to finding a very ample divisor $H$ of $S$ with $H^2 =4$. Since we know completely the intersection form on $Pic(S)$, one can try (in fact, I have been trying) to find a primitive divisor class $L$ such that

1. $L^2 = 4$,

2. there is no divisor $D$ such that $D^2 = 0$ and $D \cdot L = 1, 2$.

3. there is no divisor $E$ such that $E^2 = −2$ and $E \cdot L = 0$.

A result of Saint-Donat guarantees that $L$ or $-L$ is very ample.

My ultimate goal is to find out whether the following set is unbounded:

{$C \cdot L$ : $L$ is a very ample divisor on S with $L^2 =4$ }.

Is this set really unbounded?