The existence of primitive and sufficiently ample line bundles on K3 surfaces?  Let S be a surface and L be a line bundle on S. For any zero-dimensional closed subschemes x of S, there is natural map from global sections of L to the global sections of L restricting to x (which is a (r+1)-dimension vector space. ). For any positive integers r, the line bundle L on a surface S is called r-very ample if for any length r+1 zero-dimensional closed subschemes x of S, this map is surjective. For example, very ampleness implies 1-very ampleness. If L and K are both very ample then tensoring l copies of L and k copies of K is (k+l)-very ample. 
My question is, for every positive integer r, can you find an algebraic K3 surface with Picard number 1 such that the primitive ample line bundle is r-very ample?  Moreover, can you find infinitely many line bundles of this kind whose self-intersection numbers are all different? 
 A: 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 first comment below for an answer. 

Using the argument pointed out  by Ferreti in his first comment and taking care to
avoid the difficulty pointed out in his second comments, we can
reduce the existence of  the sought K3 surfaces to the following statement: 

There exists infinitely many integers
  k for which (2k)(2k +3) is 
  squarefree.

Start with a smooth quartic S in P(3) and let H be a hyperplane section. 
For a fixed r and k>>0,  the restriction of kH to S is r-very ample. 
Suppose S contains a line L. Then the linear system |E|=|H-L| 
defines a fibration by elliptic curves on S. 
Thus kH + E is also 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 obtain a K3 surface with Pic = Z  and  a r-very ample line-bundle of 
self-intersection 4k^2 + 6k = 2k(2k +3). If this  number is squarefree then
the line bundle is primitive. 

After googling a bit I found general results about squarefree values of polynomials  which seems to ensure the existence of infinitely many integers k for which 2k(2k +3) is squarefree.  

Edit:   I would like to know
if it is necessary to impose the number theoretical condition to obtain primitiviness. 
