Let $k$ be an *imperfect field* of char $p>0$ and 
$x \in \mathbb{P}^n_k$ be closed point of projective space.

In [this discussion][1] Qing Liu wrote that

>Over an imperfect field, a reduced point can not be contained 
in a smooth hypersurface if its residue field has "too 
high" inseparability degree. 

I'm not sure in which sense this statement it is true. 
The central point there which 
confuses me, is that it seems to suggest that the inseparability
degree of the residue field of $x$ gives a kind of 
"measure" how difficult it is to find a smooth hypersurface
$V \subset \mathbb{P}^n_k$ containing $x$.

Problems:

1) Which relevance has here the posed assumption that $x$ 
should be reduced? All points of $\mathbb{P}^n_k$ are reduced,
so I not understand why it should be extra added. Maybe there is 
some crucial relevance I not see up to now.

2) How to interpret it? It seems that Qing Liu's statement on 
high inseparabality degree of the residue field is a necessary, 
not sufficient statement 
for the property to be not contained in a smooth hypersurface, 
since there are of course points 
$x \in \mathbb{P}^n_k$ with residue field $\kappa(x)$ of arbitrary 
big inseparability degree beeing contained in a hyperplane. Or maybe it's a statement for *general* hypersurfaces containing $x$, I don't know, that's just a guess; cp. also with point 3 below)

Here a counterexample: the point $x \in \mathrm{Spec}(k[w,v]) 
\subset \mathrm{Proj}(k[W,V, U])$ (with $w:=W/U, v:=V/U$ local coords)
associated to maximal
ideal $\mathfrak{m}_x := (w^{p^n} - t, v) \subset k[w,v]$
where $t \in k$, the residue field is 
$\kappa(x) = k(t^{\frac{1}{p^n}})$, but $x$ is obviously 
contained in smooth hyperplane $(V=0)$.


3) For every point $x \in \mathbb{P}^n_k$
Bertini's theorem would give us at least a 
*regular* hypersurface containing $x$, but it is known 
that over imperfect field not every regular scheme is smooth.
But that's a binary condition: imperfect base field 
implies that regular not equivalent to smooth.

What does it have to do with *how big* the inseparability degree 
of the residue field is, and not just that it is not separable? If I'm not missing the point it 
seems that Qing Liu want to say that 
the obstruction for a regular hypersurface containing
the point $x$ to be even smooth sits in the 
inseparability degree of the residue field of $x$. But is it true
(at least as a statement for general hypersurfaces, since
point 2 gives counterexple) and how 
to see it? 


  [1]: https://mathoverflow.net/q/112486