My goal is to use a "normal Bertini" theorem (see https://link.springer.com/article/10.1007%2Fs000130050213)

More specifically, let $k$ be a field (you may assume that k is infinite but it should be irrelevent).

Let $f:X \rightarrow Y$ be a birational morphim between normal projective varieties over $k$.

I would like to show that $f$ is residually separable, that is if $x \in X$, the extension $\kappa (x)/\kappa (f(x))$ is separable.

Obviously everything happends in the exceptional locus, so we have to deal with infinite extensions.

What I'm able to show is that $\kappa(f(x))$ is algebraically closed in $\kappa(x)$ (which would at least be necessary).

To do this, pass to the completions (which are normal using Zariski main theorem) of $\mathcal{O}_{X,x}$ and $\mathcal{O}_{Y,f(x)}$. Then use Hensel's lemma.

1. Is that sufficient ? I would like to use the Mac-Lane separability criterion (see Bourbaki algebra V.15).

2. If not is there a way around ?

Any lead would be appreciated :)