Let $X$ be a projective, nodal curve, $\pi:\tilde{X} \to X$ be its normalization and $\mathcal{L}$ be an invertible sheaf on $X$. The question is: Is $\deg(\pi^*\mathcal{L})=\deg(\mathcal{L})$?
As stated in Proposition $2.1$ of this note, this seems to be the case. However, I am not able to follow the proof or the reference mentioned in this note.
Any hint/reference for this question will also be very welcome.