A divisorial valuation restricts to a divisorial valuation or the trivial valuation

By [F1], the valuation ring of $v$ is ${\mathcal{O}}_{X,{\eta }_D}$, with residue field $k(D)$ of transcendence degree $m-1$. Its intersection with $K$ is the valuation ring of $w$, whose residue field embeds in $k(D)$. If $w$ is trivial, that intersection is $K$ itself. The extension given by [F2] therefore specializes the generic point of $Y$ to the generic point under $D⇢Y$, so this map is dominant. Conversely, dominance of $D⇢Y$ implies that every nonzero rational function of $Y$ has nonzero residue in $k(D)$, hence value zero. If nontrivial, the subgroup $w(K^{\ast })\subset \mathbb{Z}$ is $d\mathbb{Z}$ for a positive integer $d$; rescaling gives a discrete valuation.

For any finite family ${\bar{h}}_1,\dots ,{\bar{h}}_t\in \kappa (v)$ algebraically independent over $\kappa (w)$, lift them to $h_i\in {\mathcal{O}}_v$. They are algebraically independent over $K$: a polynomial relation with coefficients in $K$ can be divided by a coefficient of smallest $w$-value; its coefficients then belong to ${\mathcal{O}}_w$ and at least one is a unit. Reduction gives a nonzero polynomial relation among the ${\bar{h}}_i$, a contradiction. Thus $t\le {\mathrm{trdeg}}_{K}L=m-n$. The residue fields form a tower over $k$, so [F3] gives ${\mathrm{trdeg}}_k\kappa (w)\ge n-1$. In the nontrivial case choose $t_0\in K$ of positive $w$-value. Any lifts $f_1,\dots ,f_r$ of algebraically independent residues, together with $t_0$, are algebraically independent over $k$: in a putative polynomial relation expanded in powers of $t_0$, the nonzero coefficient of the smallest power has value zero, whereas all subsequent terms have larger value. Therefore $r+1\le n$. It follows that ${\mathrm{trdeg}}_k\kappa (w)=n-1$.

In the nontrivial case choose $f_1,\dots ,f_{n-1}\in {\mathcal{O}}_w$ with algebraically independent residues and let $Z$ be the reduced closure of the graph of $Y⇢{\mathbb{P}}^{n-1}$ given by $[1:f_1:\cdots :f_{n-1}]$ (for $n=1$ take ${\mathbb{P}}^0$). The projection $Z\to Y$ is proper by [F2], and birational because it is the graph over a dense open. Normalize $Z$ to obtain $Y^{\mathrm{\prime }}$; [F4] makes this a finite proper normal modification. By [F1] the induced rational map $f^{\mathrm{\prime }}:X⇢Y^{\mathrm{\prime }}$ is defined at ${\eta }_D$. Its image closure $B$ maps dominantly to ${\mathbb{P}}^{n-1}$ because the specialized coordinates are the chosen algebraically independent residues, hence $\dim B\ge n-1$ by [F3]. The centre of $w$ cannot be the generic point of $Y^{\mathrm{\prime }}$: the local ring at that point is $K$, which is not contained in its nontrivial valuation ring. Thus $B\mathrm{\ne }{Y}^{\mathrm{\prime }}$ and $\dim B\le n-1$. Therefore $B$ is a prime divisor, and $f^{\mathrm{\prime }}{\mathrm{\mid }}_D$ dominates it. AC is inherited from the stated suppliers and the choices of transcendence bases.