Are there known examples of smooth projective hypersurface in $\mathbb{P}^3$, say $X$ and an invertible sheaf $L$ on $X$ with $H^0(X,L)>0$ satisfying the following property: There exists an infinitesimal deformation $X'$ of $X$ such that $L$ lifts as an invertible sheaf $L'$ on $X'$ (meaning $L'$ pulls back to $L$ on $X$) but there is a section $s \in H^0(L)$ which does not lift to a global section of $H^0(L')$?
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1$\begingroup$ Let $p\in S$ be a point on a smooth surface and $C\subset S$ a curve with multiplicity $m(m-1)>C^2$ at $p$. Let $f:X\to S$ be the blow up of $p$ and $L=\mathcal O _S(f^*C-mE)$ where $E$ is the exc. curve and $s\in H^0(L)$ corresponding to $f^{-1}_*C$ (such examples were constructed by Miranda; pg 279 Laz. book). If $X'$ and $L'$ are induced by deforming the point $p\in S$, then I think that $s$ does not lift to $H^0(L')$ (otherwise, following an argument of G. Xu, see eg proof of 2.1 of arxiv.org/pdf/0809.2160.pdf, we should have $C^2\geq m(m-1)$.....sorry I didn't check the details). $\endgroup$– HaconCommented Jan 19, 2016 at 17:29
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