Are degrees and ramification degrees preserved upon passing to the smooth compactification?

Question

Let $\phi ：C_1\to C_2$ be morphism of projective singular curve. Let $\tilde{C}_1$ and $\tilde{C}_2$ be their smooth compactification. Then $\phi$ extends to $\tilde{\phi} : \tilde{C}_1\to \tilde{C}_2$. Let $\deg \phi$ be degree of $\phi$ and $e_\phi (P)$ be ramification degree at $P \in C_1$.

Does $\deg \phi=\deg \tilde\phi$ and $e_{\phi} (P)=e_{\tilde \phi} (P)$ hold?

Background. If this is true, we can easily calculate the genus of $C_1$ from he genus of $C_2$ by applying the Riemann–Hurwitz formula to $\tilde{\phi}$.

N.B. Sorry, my link was wrong, the meaning of smooth compactification is in this paper of page $2$, https://arxiv.org/pdf/2112.02470.pdf

Answer 1

The first claim is true. The degree can be defined as the degree of the extension on function fields $k(C_1)/k(C_2)$, but $k(\tilde{C}_1) = k(C_1)$ and $k(\tilde{C}_2)= k(C_2)$ so $k(C_1)/k(C_2) = k(\tilde{C}_1)/k(\tilde{C}_2)$.

The second claim is true for smooth points lying over smooth points. This is because it is clear from the definition of ramification degree that it may be computed locally (say in the Zariski topology) and $C_1$ is isomorphic to $\tilde{C}_1$ locally in the Zariski topology around smooth points. I am not sure if one can define the ramification degree for singular points.

We would normally refer to what you call the "smooth compactification" as the "normalization" or maybe "resolution of singularities" since it's not the compactification of $C$ but rather of a smooth open subset of $C$.