Arithmetic deformation of line bundles

Abstract We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for families $$f : \mathcal {X} \rightarrow \mathcal {S}$$ f : X → S with big monodromy and large period image defined over the ring of N -integers $$\mathcal {O}_{L}[1/N]$$ O L [ 1 / N ] of a number field L , we produce a proper closed subscheme $$\mathcal {E} \subsetneq \mathcal {S}$$ E ⊊ 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 $$\mathbb {P}^1$$ P 1 and projective hypersurfaces in $$\mathbb {P}^3$$ P 3 of degree $$d \ge 5$$ d ≥ 5 . We also study the locus $$\mathcal {E}$$ E in more detail in the $$h^{0, 2} = 2$$ h 0 , 2 = 2 case.