Arithmetic deformation of line bundles

We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for families f : X → S f : X → S with big monodromy and large period image defined over the ring of N -integers OL[1/N] O L [ 1 / N ] of a number field L , we produce a proper closed subscheme E 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 P¹ P 1 and projective hypersurfaces in P³ P 3 of degree d ≥ 5 d ≥ 5 . We also study the locus E E in more detail in the h0, 2 = 2 h 0 , 2 = 2 case.