Upper semi-continuity of intersection numbers

Question

Consider a smooth projective morphism of schemes $X \rightarrow S$ with relative dimension $n$ (the application I have in mind is with $S$ = an open subset of $\text{Spec } \mathbb{Z}$) and assume that $V$ and $W$ are two closed subscheme of $X$, flat over $S$, such that $\text{codim}(V) + \text{codim}(W) = n$.

Denote the fibers of $X,V$ and $W$ over $s \in S$ by $X_s,V_s$ and $W_s$ and write $A^{\bullet}(X_s)$ for the Chow ring of $X_s$. The proper pushforward of the structure morphism of $X_s$ induces a morphism $\text{deg} : A^{n}(X_s) \rightarrow \mathbb{Z}$.

For every $s \in S$, the intersection of $V_s$ and $W_s$ in $X_s$ gives an intersection number $\text{deg}([V_s].[W_s]) \in \mathbb{Z}$. Is there an upper semi-continuity result on the function $s \mapsto \text{deg}([V_s].[W_s]) \in \mathbb{R}$ ?

My question is related to this post : Semi-continuity of intersection numbers but the example given in the comments does not satisfy the flatness assumption.

Answer 1

In fact the intersection numbers are constant: by [Fulton, Cor. 20.3], the specialisation maps $\sigma \colon A^*(X_\eta) \to A^*(X_s)$ are ring homomorphisms if $S$ is a Dedekind scheme with generic point $\eta$ and closed point $s$. By [Fulton, §20.2] the intersection product for flat cycles is just the fibre product: $$[V] \underset S\times [W] = \left[ V \underset S\times W \right].$$ Finally, degree of $0$-cycles is preserved under specialisation (use that a finite flat $S$-scheme is locally free). Thus, we can compare $\deg([V_s] \cdot [W_s])$ to $\deg([V_{s'}] \cdot [W_{s'}])$ by identifying both with $\deg([V_\eta] \cdot [W_\eta])$. $\square$

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References.

[Fulton] W. Fulton, Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 2. Berlin: Springer (1998). ZBL0885.14002.