As in the picture, I have a question on the proof of part (b).I think I have found an effective divisor $D=\sum P_i$ on $\widetilde X$ with $L(D)\cong L$ and such that $f(P_i)$ is nonsingular point on X, how can I conclude there is an invertible sheaf $L_0$ on $X$ with $f^*L_0 \cong L $ from this? Can anyone give some hints?
A picture to have in mind: if $X$ is a nodal cubic curve $y^2=x^3+x^2$, its normalization is just separating the two collapsed points at the origin (imagine that the singular point is an overlapping of two points).
Normalization only modifies the singular point. In particular, $L$ is the pullback of the line bundle corresponding to the image of the divisor. Normalization is finite surjective. For a finite surjective morphism, a line bundle at the target is ample iff its pullback is ample. (see Ex 5.7d).