This is not an answer but instead related questions.
In Beauville's "Counting rational curves on K3 surfaces" is implictly assumed the existence of algebraic K3 surfaces with Pic of rank one and generated by a curve of genus g.
How do we show the existence of such K3 surfaces ? Edit: See Ferreti's comment below for an answer.
Do these K3 surfaces (for g>>0) answer the question ?