Arithmetic Deformation of Line Bundles
David Urbanik, Ziquan Yang
TL;DR
This work develops a framework to study mixed-characteristic liftability of line bundles in large arithmetic families $f: \mathcal{X}\to\mathcal{S}$ over $\mathcal{O}_L[1/N]$ by combining integral period-jet techniques with Ax–Schanuel-type results. It identifies a proper closed locus $\mathcal{E}\subsetneq\mathcal{S}$ such that, outside $\mathcal{E}$, the relative Picard scheme is syntomic and every arithmetic Noether–Lefschetz component has the expected codimension $h^{0,2}$, with line bundles in positive characteristic lifting to characteristic zero via specialization data. The paper gives concrete applications to elliptic surfaces and degree $d\ge 5$ hypersurfaces in $\mathbb{P}^3$, and provides a detailed $h^{0,2}=2$ analysis. The methods transfer Hodge-theoretic unlikely-intersection ideas to the integral setting, producing an explicit, computable structure for liftability and a pathway to constructing liftings or rational points from liftable line bundles. Overall, the results bridge transcendental period techniques with arithmetic deformation theory to control when line bundles on reductions lift to characteristic $0$.
Abstract
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.