Blow up $\operatorname{Pic}^3(C)$ along $C$.

Question

I want to solve this problem

from E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris (auth.) - "Geometry of Algebraic Curves Volume I"

I can show that the fibers of $\{(K_C-P) \in \operatorname{Pic}^3(C) | P\in C\}$ are $\mathbb{P}^1$ using R-R theorem. For other points $u$ is isomorphism. But I can't understand why this map should be blow up. Probably there are some criteria (e.g. Castelnuovo's Contractibility Theorem but for higher dimension). Than we can do something similar to the first answer here.

Thanks