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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 :)

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    $\begingroup$ New contributor, welcome! Take $Y=\mathbb{P}^2$, and $f=$ the bolwing-up at a rational point $y$. Then $f^{-1}(y)\cong \mathbb{P}^1$ which contains plenty of points with inseparable residue field. $\endgroup$
    – Laurent Moret-Bailly
    Commented Jul 28, 2019 at 5:49
  • $\begingroup$ Ok thank you! I wasn't loking for something this simple because I thought my idea worked on closed points. I will have to find a way around it then. $\endgroup$
    – Carot
    Commented Jul 28, 2019 at 7:36

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