# Upper semi-continuity of intersection numbers

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.

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 $$\deg([V_\eta] \cdot [W_\eta])$$. $$\square$$
• I understand your answer in the case where $S$ is the spectrum of a discrete valuation ring. For a Dedekind scheme, the specialization map in Fulton is a map from the Chow ring of the fiber of $X$ over $S-s$ and not over the generic point. How do you get your result from this ? Thank you – user158892 Jun 4 '20 at 8:28
• Apply the above to each DVR $\mathcal O_{S,s} \subseteq \mathcal O_{S,\eta}$. – R. van Dobben de Bruyn Jun 4 '20 at 22:09