Let $\sigma$ be a maximal cone of $\Sigma$. We denote by $\mathscr{L}^{\sigma}_{j}$ and $\mathscr{L}^{\sigma}_{i}$ the line bundles on $\mathfrak{X}$ induced respectively by $\mathcal{O}_{\mathscr{X}}(W^{\sigma}_{j})$ for $j\in J$, and by $\mathcal{O}_{\mathscr{X}}(W^{\sigma}_{i})$ for $i\in L_{\sigma}$. Since $W^{\sigma}_{j}$ and $W^{\sigma}_{i}$ are principal on $Z$, the restrictions ${\mathscr{L}^{\sigma}_{j}}_{|Z}$ and ${\mathscr{L}^{\sigma}_{i}}_{|Z}$ are trivial line bundles on $Z$, thus we may choose non-zero global sections $s^{\sigma}_{j}$ and $s^{\sigma}_{i}$ on $Z$.

We now lift the sections $s^{\sigma}_{j}$ and $s^{\sigma}_{i}$ to global sections of $\mathscr{L}^{\sigma}_{j}$ and $\mathscr{L}^{\sigma}_{i}$, which we still denote by $s^{\sigma}_{j}$ and $s^{\sigma}_{i}$. Indeed, for any $n\geqslant 1$, write $(\mathscr{X}/Z)_{n}$ for the (non-reduced) subscheme of $\mathscr{X}$ defined by the ideal $\mathscr{I}^{n}_{Z}$. In the exact sequence

and in the analogous one for $\mathscr{L}^{\sigma}_{i}$, the right-hand vanishes: the conormal bundle is a direct sum of line bundles on $Z$ which are nef by the hypothesis in Theorem B and so are its positive tensor powers, thus their first cohomology group vanishes by Proposition 1.2.5. We thus extend the sections constructed above to all of the $(\mathscr{X}/Z)_{n}$ by induction, which yields an extension to $\mathfrak{X}=\varprojlim_{n}(\mathscr{X}/Z)_{n}$.