Let $S$ be the spectrum of a discrete valuation ring with generic point $\eta$. Let $C/\eta$ be a smooth connected curve with an $\eta$-valued point, and let $\mathcal{C}/S$ be the smooth locus of the minimal proper regular model of $C$ over $S$. Let $N/S$ denote the Neron model of the Jacobian of $C_\eta$, and let $\alpha:C \rightarrow N_\eta$ denote the Abel-Jacobi map.

Now by the Neron mapping property, we obtain a (unique) extension of $\alpha$ to $\overline{\alpha}:\mathcal{C} \rightarrow N$.

Question: Is the map $\overline{\alpha}$ necessarily an immersion?

If the genus is 1, we are OK! In general, I cannot think of a reason why this should be true, but I also cannot think of a counterexample.

I started thinking about this question from the point of view of Serre's book `Algebraic groups and class fields', ie as a question about line bundles on singular curves. I wanted to use such a result in my thesis, but in the end found an alternative approach. However, I am still curious...

Thank you for your time.