In David Mumford's book Algebraic Geometry I, Complex Projective Varieties the
proof of (3.25) **Specialization principle** on page 53 contains an argument
I not understand.


General assumptions: all our varieties are over $\mathbb{C}$. The statement is:



> 
> (3.25) **Specialization principle.** Let $Z \subset \mathbb{P}^n \times \mathbb{P}^m $
> be $r$-dimensionally subvariety and $X =p\_1(Z) \subset \mathbb{P}^n$ for
> $p\_1: \mathbb{P}^n \times \mathbb{P}^m \to \mathbb{P}^n $. Suppose
> $\operatorname{dim} X=r $ and let
> 
> 
> $$\phi= \text{ res } p\_1: Z \to X $$
> 
> 
> is almost everywhere (= on an open set) finite to one. [... ]
> 
> 
> Then the map
> 
> 
> $$ F: X\_1 \to \mathbb{N}, x \mapsto \# \phi^{-1}(x)$$
> 
> 
> where
> $X\_1 := \{x \in X \ \vert \ X \text{ smooth at } x \text{ and } \phi^{-1}(x) 
> \text{ finite } \}$ is *lower semi-continuous* in the *Zariski* topology on $X\_1$.
> 
> 
> 


Lower semi-continuous means that for every $n \in \mathbb{N}$ the set
$\{x \in X \ \vert \ \# \phi^{-1}(x) \ge n \}$ is closed.


The first part of the proof shows that $ F: X\_1 \to \mathbb{N}$ is
*lower semi-continuous* in the *classical* topology, recall a smooth complex
variety can be canonically endowed with classical analytical topology
considering it as a complex manifold.


Mumford remarks that we can observe that for every $n \in \mathbb{N}$ the set
$\{x \in X \ \vert \ \# \phi^{-1}(x) \ge n \}$ is *constructible*, ie a union of finite
number of locally closed sets in Zariski topology.


Then Mumford claims that this implies that $F$ is
*lower semi-continuous* in the *Zariski* topology.


That's the point I not understand. Does anybody see
why the implication $F$ lower semi-continuous with resp classical topology
and that $\{x \in X \ \vert \ \# \phi^{-1}(x) \ge n \}$ is *constructible* imply
$F$ lower semi-continuous with resp Zariski topology?



Comparison of classical and Zariski topologies with constructible sets
First, for lower semi-continuity, we should consider sublevel set, i.e.
$$\{x\in X:\ \#\phi^{-1}(x)\leq n\}$$
Now, since $\#\phi^{-1}(x)$ is a constructible function, the above subset, denoted as A, satisfies that
\begin{equation}
A=\bigcup\_{i=1}^N (Z\_{i,1}-Z\_{i,2})
\end{equation}
where $Z\_{i,j}$'s are all Zariski closed subsets. Also, from the lower semi-continuity of $\#\phi^{-1}(x)$ in the classical topology, A is closed in the classical topology. Taking the classical closure of the above equation on $A$, we get
$$A=\bigcup\_{i=1}^N Z\_{i,1}$$
which is Zariski closed. This shows that $\#\phi^{-1}(x)$ is also lower semi-continuous in the Zariski topology.


