Arithmetic Deformation of Line Bundles

Abstract

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 ${\mathcal{O}}_L[1/N]$ 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.

38 pages, minor revisions

References in corpus (7)

• Néron-Severi groups under specialization
• Bertini irreducibility theorems over finite fields
• On the Tate and Mumford-Tate conjectures in codimension one for varieties with h^{2,0}=1
• Computing Néron-Severi groups and cycle class groups
• Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields
• Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results
• Sets of Special Subvarieties of Bounded Degree