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
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.