are they correct at all ? I am willing to assume everything lies a very nice ambient space, say

holomorphically convex manifold.

\def\hatZ{{\hat Z}}

Let $\nnn:\hatZ \ra Z$ be the normalisation of a variety $Z$, and

let $\Zz k(\C) = \{z\in Z(\C): \#\nnn\inv(z)\geq k\}$ be a closed

subvariety of points which have $k$ preimages or more.

\begin{fact}\label{fact:zknrm} $\Zz k$ is a closed subvariety of $Z$, for all $k$;

$\Zz {\deg \nnn + 1}$ is empty, $\Zz 1 = Z$. \end{fact} \bp

Reference!!!! \ep

\begin{fact}\label{fact:214} Around each point $z\in Z(\C)$ there exists a

sufficiently small neighbourhood $z\in V$ open in complex topology

such that

$$V\cap \Zz k= \bigcup\limits_{i_1<..<i_k} V_{i_1}\cap ...\cap V_{i_k},$$ where $$Z\cap V=V_1\cup ...\cup V_n$$ is the irreducible decomposition of $Z\cap V$.\end{fact} \bp not [Grauert, ???] [Abhyankar, Local Analytic Geometry, p.402, Cl 7] \ep

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