Comparison of classical and Zariski topologies with constructible sets

Question

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?

Answer 1

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.