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. 

---

**Edit**: Using the argument pointed out below by Ferreti, we can
prove the existence of the sought K3 surfaces. Start with a 
smooth quartic S in P(3). For a fixed r and k>>0,  the restriction 
of O(k) to S is r-very ample. Let SS be a family of K3 surfaces 
that  deforms S in such a way that the class of O(k) is preserved, 
and for a generic member of the family every  element in H1,1\cap
H^2(Z) not proportional to O(k) becomes non-rational. Thus the 
generic element has Pic = Z. Since r-very ampleness is an open condition
( the points in the relative Hilb^r(SS)  where it does not hold is closed)
we have a positive answer for the first question.

---

I am not sure if I understood correctly the second question. Aren't
 the positive powers of every r-very ample line-bundle  also 
r-very ample ?