I have following question about so-called "principle of degeneration" 
in algebraic geometry (which in modern terms is an immediate consequence
of Zariski's main theorem and goes in it's [original form](https://en.wikipedia.org/wiki/Zariski%27s_connectedness_theorem) to 
Enriques) saying roughly that for a (wlog local) normal base ring $R$
with field of fractions $K:=\text{Frac}(R)$ a (geometrically) connected, pure $r$-dimensional projective subscheme 
$X_K \subset \mathbb{P}_K^n$ whose vanishing ideal $I(X)$ can be generated
by polynomials in $R[X_0,..., X_n]$ specializes to a connected subscheme 
over a special point $s \in \operatorname{Spec}(R)$.

In modern terms it should be phrased as follows: that for any point
$s \in \operatorname{Spec}(R)$ specializing the generic point of $R$ - so every point/prime $\mathfrak{p}_s \subset R$ - the fiber 
$X_s =X \otimes_R \kappa(s)$ over $\kappa(s)$ is connected.
(note, that by assumption $X \subset \mathbb{P}_R^n$ (ie as scheme over
$R$ make sense since we assumed that it's vanishing ideal $I(X)$ is generated by
polynomials in $R[X_0,..., X_n]$)

Now the **question:** Wei-Liang Chow remarked in his paper [On the connectedness theorem in algebraic geometry][1] briefly (at first page) that even though it is hard to prove 
pure algebraically (indeed Zariski invented to show this rigorously
whole apparatus of formal holomorphic functions)
in contrast it is *rather easy* to show this in classical 
(=complex analytic) context by "transcendental methods".

Could somebody sketch the rough idea how to show 
this result on "Principle of Degeneration" with complex analytic methods 
what should be due to Chow not so hard?
(indeed the notion of normality exists clearly also in analytic world,
so the normal base ring $R$ makes also sense in complex analytic
setting) 

Why is this interesting (for me): I would like to develop better 
"geometric" intuition for nature of normality as necessary condition in this context.

In modern terms normality of a scheme/ring can be formulated in pure 
category theoretic terms as a universal property to be the 
"maximal" finite birational extension in the sense of if that if $X$ is normal
then if $f:Y \to X$ is finite birational, then it's already an iso
(modern argument: immediately by Stein factorization).  

But I would like to go back in time a bit to unravel some black boxes
to understand better the "inner mechanism" why it is necessary that the base ring is normal in the sense that it contains it's all "integral" elements.

This suggests, that somehow the whole claim could be somehow reduced to approval that a solution on a monomial equation with coefficients in $R$ which lives in $K$ already "lives" in $R$  (essentially that's what $R$ normal means).

Let consider following "toy situation": $R$ as before normal ring, 
$\mathfrak{p} \subset R$ a prime, and a geometrically connected 
$X_K =V_+(F)  \subset \mathbb{P}_K^n$ is given by a simple homogeneous 
polynomial $F(X_0,..., X_n) \in R[X_0,..., X_n]$ (ie coeff in $R$!)

RMK on geometrically connectedness: One way to assume it is to assume
that $X_K$ is connected and $K$ is separably closed. In analytic situation
$K$ contains $\mathbb{C}$, let's do it, as I'm primary interested in "quick"
analytic method Chow skipped in the linked paper.

And the question is if there is a direct argument based strongly on
assumed normality of ring $R$ (resp it's localization 
at $\mathfrak{p}$)
to see that 
$X_{\kappa(\mathfrak{p})} = V_+(\overline{F}) 
\subset \mathbb{P}_{\kappa(\mathfrak{p})}^n$ (here $F$ with coefficients
reduced modulo $\mathfrak{p}$ (+localization) must be also connected.

Is there an ad "heuristic" argument why it is the case? (at leat, why it is plausible to expect
?)

  [1]: https://www.jstor.org/stable/1970117?saml_data=eyJzYW1sVG9rZW4iOiJmNGE1MDY1OS1jM2I0LTRhM2ItYTNhNy1hZTQyOTkyYzhiMWUiLCJpbnN0aXR1dGlvbklkcyI6WyJkYzNjZThmNC0yNzkxLTQxMjYtYTA2ZS1jNTUxNjdiZDE0NzciXX0