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](https://mathoverflow.net/questions/238086/semi-continuity-of-intersection-numbers) but the example given in the comments does not satisfy the flatness assumption.



Upper semi-continuity of intersection numbers
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](https://zbmath.org/?q=an:0885.14002).


