Arithmetic Deformation of Line Bundles

We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for sufficiently large smooth projective families $f:\mathcal{X}\to \mathcal{S}$ defined over the ring of $N$-integers ${\mathcal{O}}_L[1\mathrm{/}N]$ of a number field $L$, we produce a proper closed subscheme $\mathcal{E}⊊\mathcal{S}$ outside of which all line bundles appearing in positive characteristic fibres of $f$ admit characteristic zero lifts. This in particular applies to elliptic surfaces over ${\mathbb{P}}^1$ and projective hypersurfaces in ${\mathbb{P}}^3$ of degree $d\ge 5$. We also study the locus in $\mathcal{E}$ in more detail in the $h^{0,2}=2$ case.