Lifting a linear surface from a curve to the ambient surface

Let $$X$$ be a complex K3 surface and $$C$$ a smooth curve on $$X$$ and $$A$$ a basepoint free line bundle on $$C$$.

Aprodu's paper - Lazarsfeld Mukai bundles and applications says this. We cannot lift the linear system $$|A|$$ to $$X$$ if $$Pic\,X$$ is generated by $$C$$ or if $$X$$ contains no elliptic curves, for most $$|A|$$.

I do not understand this statement. I know that in general $$A$$ need not be a restriction of a line bundle from $$X$$. But he seems to be saying more. Any clarification of the above statement will be helpful.

• I do not completely understand this statement. What if $\mathrm{Pic}(X)$ is generated by a very ample line bundle $C$ and $A=\mathcal{O}_X(C)|_C$? In this case $A$ can be clearly lifted to $X$ , one lifting being $\mathcal{O}_X(C)$. Or am I missing something? – Francesco Polizzi Jun 12 at 12:52
• @ Polizzi, of course you are correct. Omitted from the quote OP shared is 'for most |A|" . I think your comment pretty much explains what Aprodu is trying to say. – aginensky Jun 12 at 14:25
• @aginensky, What about when he says if $X$ contains no elliptic curve. I did not understand that bit? – user52991 Jun 13 at 3:11
• I have edited the question to include for most $|A|$. – user52991 Jun 13 at 3:13
• @user52991 No, that can never happen as K-3 is regular ! I don't want to speak to Aprodu, but the gyst of his comment should be that curves have way more special linear systems than most K-3 surfaces have. – aginensky Jun 17 at 13:41