Thank you! I appreciate it.

@danneks, Hi, I'm not too sure I follow; firstly, I think on $\mathcal{O}_x$, you've lost so much information that I'm not quite sure how you can conclude? Secondly, to me the statement $I_x/ f\otimes A / \mathfrak{m} = 0$ doesn't seem quite helpful, as all it's saying is that on the fibre that $x$ is in, say $X_p$, the ideal of $I_x$ in the local ring of $x$ in $X_p$, say $\mathcal{O}_{X_p, x}$ is principally generated, which, if $X_p$ is nonsingular, already follows since the local ring $\mathcal{O}_{X_p, x}$ is a DVR?

Thank you so much, that was very helpful!

Thank you! Yes, the curves I'm working with are in fact proper (I edited the question). Thank you so much!