# How does the line bundles look like on a proper model (or Néron model) of an abelian variety?

How does the line bundles look like on a proper model (or Néron model) of an abelian variety?

In particular, let us work over a trait $S=\mathrm{Spec} R$, where $R$ is a DVR, with generic point $\eta=\mathrm{Spec}K$ and special point $s=\mathrm{Spec}k$. Given an semistable abelian variety $A_K$ over $K$, we have the Néron model $A$ over $S$, and other proper models over $S$, take one proper model say $E$. Given a line bundle $L$ on $A$ or $E$, they restricts to a line bundle on $A_K$. Now when a line bundle on $A_K$ come from a line bundle on $A$ or $E$. In other words, how to understand the map $\underline{\mathrm{Pic}}_S(A)(V)\rightarrow \underline{\mathrm{Pic}}_K(A_K)(V_K)$ for $V$ some $S$-scheme, and $\underline{\mathrm{Pic}}_S(E)(V)\rightarrow \underline{\mathrm{Pic}}_K(A_K)(V_K)$, injective? surjective? image?......

Also about the map $\underline{\mathrm{Pic}}_S^0(A)\rightarrow \underline{\mathrm{Pic}}_K^0(A_K)$. In this case, for $V$ smooth over $S$, one probably can get some information from $\underline{\mathrm{Pic}}_K^0(A_K)(V_K)=A_K^*(V_K)=A^*(V)$

• Heer, could you give me an example of the kinds of things you would like to know (the kinds of things you have in mind when you say "look like")? – LMN Dec 15 '12 at 20:35
• If $\mathcal{A}$ denotes a model for $A$ over some appropriate base, there is a homomorphism $\mathrm{Pic}(\mathcal{A}) \to \mathrm{Pic}(A)$ given by restricting line bundles to the generic fibre. Is this what you are interested in? – Daniel Loughran Dec 16 '12 at 10:28
• @LMN @Daniel Loughran : sorry for this ambiguious question. I have edited my question. – Heer Dec 16 '12 at 16:36
• A Néron model is rarely proper. – Damian Rössler Dec 16 '12 at 22:18
• @Damian Rössler : yes, but I would like to know the answer for bother Neron models and proper models – Heer Dec 17 '12 at 13:40