Let $f:C\to S$ be a regular fibered surface where $S=Spec(R)$, $R=dvr$. Assume $C$ has smooth geometrically integral generic fibre $C_K$. We also assume the existence of a section $x\in C(S)$. Let $J_K$ be the Jacobian of $C_K$ and $u_K:C_K\hookrightarrow J_K$ the AbelJacobi map sending $x_K$ to $0_{J_K}$. Let $N$ be the NĂ©ron model of $J_K$. Can we find an AbelJacobi map for $C$, i.e. a morphism $u:C\to N$ extending $u_K$? We also assume $C$ is not smooth! What if $C$ is only normal? Thank you!
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