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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.

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    $\begingroup$ 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? $\endgroup$
    – Francesco Polizzi
    Commented Jun 12, 2019 at 12:52
  • $\begingroup$ @ 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. $\endgroup$
    – meh
    Commented Jun 12, 2019 at 14:25
  • $\begingroup$ @aginensky, What about when he says if $X$ contains no elliptic curve. I did not understand that bit? $\endgroup$
    – user52991
    Commented Jun 13, 2019 at 3:11
  • $\begingroup$ I have edited the question to include for most $|A|$. $\endgroup$
    – user52991
    Commented Jun 13, 2019 at 3:13
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    $\begingroup$ @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. $\endgroup$
    – meh
    Commented Jun 17, 2019 at 13:41

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