Let $R$ be a complete discrete valuation ring and let $S = \operatorname{Spec}(R)$. Let $\mathcal{X}\to S$ be a complete, regular, flat, connected $S$-scheme of finite type whose fibres are smooth, projective, geometrically connected algebraic curves. Let $s \in S$ be the closed point, let $\xi \in S$ be the generic point, let $\mathcal{X}_s$ be the special fibre of $\mathcal{X}$, and let $X = \mathcal{X}_\xi$ be the generic fibre of $\mathcal{X}$.
If $\mathscr{L}$ is an invertible sheaf on $\mathcal{X}$, then there are maps $\Gamma(\mathcal{X}, \mathscr{L})\to \Gamma(X, \mathcal{L}_\xi)$ and $\Gamma(\mathcal{X}, \mathscr{L})\to \Gamma(\mathcal{X}_s, \mathscr{L}_s)$. In the right circumstances (I think when $\mathscr{L}$ is generated by global sections; maybe we need projective normality?), these two maps induce surjections $\Gamma(\mathcal{X}, \mathscr{L}) \otimes_R k(\xi) \to \Gamma(X, \mathcal{L}_\xi)$ and $\Gamma(\mathcal{X}, \mathscr{L}) \otimes_R k(s) \to \Gamma(\mathcal{X}_s, \mathscr{L}_s)$.
My question involves going in the other direction:
Suppose we are given $\mathscr{L}_s$ on $\mathcal{X}_s$ which is generated by global sections. Under what conditions, and how, can one find an invertible sheaf $\mathscr{L}$ on $\mathcal{X}$ such that $\Gamma(\mathcal{X}, \mathscr{L}) \otimes_R k(\xi) \to \Gamma(X, \mathcal{L}_\xi)$ and $\Gamma(\mathcal{X}, \mathscr{L}) \otimes_R k(s) \to \Gamma(\mathcal{X}_s, \mathscr{L}_s)$ are surjective.
ETA: Note that I am particularly interested in the "and how" part; which is to say that, given an (explicit) basis for $\Gamma(\mathcal{X}_s, \mathscr{L}_s)$, how do I go about finding an (explicit) basis for $\Gamma(\mathcal{X}, \mathscr{L})$ that satisfies the surjectivity properties above (assuming $\mathscr{L}$ exists)?
My idea is to take a basis of $\Gamma(\mathcal{X}_s, \mathscr{L}_s)$, lift it to an $R$-module $M$, and take the saturation $\operatorname{Sat}(M) = R^n \cap (M \otimes_R k(\xi))$ where $n = \dim\Gamma(\mathcal{X}_s, \mathscr{L}_s)$. Some related "sub-questions" based on this idea:
- Suppose $\Gamma(\mathcal{X}_s, \mathscr{L}_s)$ is generated by global sections $\{\bar{s}_0, \ldots, \bar{s}_r\}$ and let $M$ be the $R$-module generated by lifts $\{s_0, \ldots, s_r\}$ of the $\bar{s}_i$. Let $\mathscr{M} = \widetilde{M}$ be the sheaf associated to $M$. Then $M = \Gamma(\mathcal{X}, \mathscr{M}) \to \Gamma(\mathcal{X}_s, \mathscr{L}_s)$ is surjective. Under what conditions is $\mathscr{M}$ invertible?
- Is this $\mathscr{M}$ isomorphic to $\pi_*\mathscr{L}_s$, where $\pi\colon\mathcal{X}_s \to \mathcal{X}$ is the canonical "projection"?
N.B. The conditions I listed in the first paragraph describe the (rather restrictive) situation I'm interested in; I would certainly be interested in hearing answers valid in a more general context. Feel free to strengthen any conditions as necessary; for example, I think projective normality might play a role somewhere, but I'm not sure where exactly.