Abstract 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 O₋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 h^0, 2 = 2 h 0, 2 = 2 case.
Urbanik et al. (Thu,) studied this question.
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