Existence of curve nodal at given set of points

Question

I am reading this (https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/Univcodim2invent.pdf) paper of Voisin, but I am having some trouble with the proof of Sublemma 2.8 (it might be something very simple but I don't exactly see it). Namely, let $S$ be a $k\leq 7$ nodal quartic surface in $\mathbb{P}^3$ and let $Z$ be the set of nodes. Let $\pi:\tilde{S}\longrightarrow S$ be the blowup of $S$ along $Z$. Then one gets that the "principalized" linear series $|\pi^\ast I_Z(4)|$ is nef and big on $\tilde{S}$ (here $I_Z\subset O_S$ is the ideal sheaf of theset $Z$). From here, how can one deduce that there exists an irreducible curve in $|I_Z(4)|$ on $S$ which is nodal at $Z$? In particular is the bigness hypothesis necessary?
I know how to obtain an irreducible curve through $Z$ by taking hyperplane sections on $\tilde{S}$ and pushing it down to $S$. However how do I know that the image will be nodal? I think this should follow from the fact that the singularities of $S$ are nodes but how to show that properly?

Answer 1

The curve upstairs (hyperplane section) is smooth I think so its projection (the curve downstairs) is evidently resolved by a single blowup along $Z$. So I believe that makes it nodal --- non-nodal singularities would require more blowups.