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

`\tilde{C_1}`

to $\tilde{C}_1$`\tilde{C}_1`

, which I think looks better; but, if you do prefer the tilde to be placed over everything, then perhaps $\widetilde{C_1}$`\widetilde{C_1}`

looks better. Finally, I had trouble understanding your "cf.", which I would usually take to mean "compare". Is that what is meant here? $\endgroup$