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I will fix notation: $\Delta = \mathrm{Spec} R$ denotes a discrete valuation ring and $\Delta^*=\mathrm{Spec} K$ for $K=\mathrm{Frac}(R)$. Suppose we are given a curve $\pi:C\to \Delta$ and a line bundle $\mathcal{L}$ on $C$. Let $U = \pi^{-1}(\Delta^*)$. If I am given a section $s \in \mathcal{L}(U)$, are there any results that help me to extend this $s$ over the central fibre $C_0$ (the fibre over the closed point of $\Delta$)?

If I assume that the generic fibre is irreducible, the whole family is irreducible and our section $s$ defines a rational section $\mathcal{L}$. However, I would like to obtain a value for every point in the central fibre, possibly infinite, if it is possible. For instance, if we regard a meromorphic/rational section of $\mathcal{L}$ as being a section of $\Bbb{P}(\mathcal{L}\oplus \mathcal{O}_C)$, then we can extend the section over codimension $1$ points by properness, but this only extends $s$ to the generic point of the central fibre.

Briefly, what tools, if any, are available for extending a $s\in\mathcal{L}(U)$ to the central fibre, where we view $s$ as a section of $\Bbb{P}(\mathcal{L}\oplus \mathcal{O}_C)$?

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    $\begingroup$ A section over a dense Zariski open $U\subseteq C$ is a quotient coherent sheaf of $\mathcal{L}_U \oplus \mathcal{O}_U$ that is an invertible sheaf on $U$. There is a unique extension to a quotient coherent sheaf of $\mathcal{L}_C\oplus \mathcal{O}_C$, but that extension need not be invertible. Consider the case that $R = k[t]_{\langle t \rangle} \subset k(t)=K$ and $C$ equals $\text{Spec}\ R[s]$. Let $\mathcal{L}$ be $\mathcal{O}_C$. Consider the surjection from $\mathcal{O}_C\oplus \mathcal{O}_C$ to the (maximal) ideal sheaf $\langle t,s\rangle$. Altman-Kleiman study this by blowing up. $\endgroup$
    – Jason Starr
    Commented Jun 15, 2022 at 11:09
  • $\begingroup$ Thanks for the pointer - in which of the Altman-Kleiman articles is this (if you happen to know)? $\endgroup$
    – Alekos Robotis
    Commented Jun 15, 2022 at 14:23
  • $\begingroup$ I am thinking of the articles "Compactifying the Picard scheme" and "Compactifying the Picard scheme, II." $\endgroup$
    – Jason Starr
    Commented Jun 15, 2022 at 14:40
  • $\begingroup$ Thanks. I've taken a cursory glance through the articles and haven't quite found what I want yet. Is it the case that the extended quotient coherent sheaf needs to be torsion free in general? If this is the case then I think I might have an idea what to do. Sorry for all the questions. $\endgroup$
    – Alekos Robotis
    Commented Jun 15, 2022 at 21:10
  • $\begingroup$ Yes, the extended coherent quotient sheaf is torsion-free. $\endgroup$
    – Jason Starr
    Commented Jun 15, 2022 at 23:03

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