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 $f : \mathscr{X} \to \mathscr{S}$ defined over the ring of $N$-integers $\mathscr{O}_{L}[1/N]$ of a number field $L$, we produce a proper closed subscheme $\mathscr{E} \subsetneq \mathscr{S}$ 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 $d \geq 5$. We also study the locus in $\mathscr{E}$ in more detail in the $h^{0, 2} = 2$ case.