# Codimension restrictions on intersections

This is a question I stumbled across earlier this week. I see a similar one has been asked here.

Suppose I have a smooth quasi projective variety $X$ over a field $K$, and I call $\text{Chow}^r(X\times \mathbb{A}^n_K)$ the Chow variety of effective cycles of codimension $r$ in $X\times\mathbb{A}_K^n$.

Now let's pick a closed subvariety $Z\subset X\times\mathbb{A}^n_K$ of the form $X\times D$ for $D$ an effective divisor on $\mathbb{A}^n_K$, and define

$$\text{Chow}^r(X)(Z)$$ to be the subset of $\text{Chow}^r(X)$ of the cycles whose supports intersect $Z$ in codimension higher than $k$, say ($k$ fixed). Here we assume the dimension of $X$, $n$, $r$ and $k$ are consistent.

Is $\text{Chow}^r(X)(Z)\subset \text{Chow}^r(X)$ open?

When is a restriction on the codimension of intersections with a fixed closed subvariety (e.g. codimension $\ge k$) an open condition on $\text{Chow}^r(X)$?

• If $X$ is not proper, there typically is no "Chow variety of effective cycles" in $X$. The Chow functor (say, on the category of seminormal, separated, finite type $K$-schemes as in the book Rational curves on algebraic varieties by Koll'ar) is not representable. You could just ask your question for some particular family of cycles over a specified base scheme. However, the answer is negative for the same reason as in the other comment that I wrote today. Jan 14 '18 at 15:22
• @JasonStarr Upon assuming the characteristic of $k$ is zero, Kollar's Chow functor is representable in the quasi projective case too (Thm. 5.4 loc cit). I agree with your argument about the non openness aspect.
– user113452
Jan 14 '18 at 21:26
• Koll'ar's Chow functor is defined to be the functor of effective cycles that are also proper over the base. Jan 14 '18 at 21:32
• If you have a family of cycles that is proper over the base, then the intersection of that family of cycles with the closed subset $Z$ is also proper over the base. Then you can apply Chevalley's theorems about upper semicontinuity of fiber dimension to prove openness of the locus in the base over which the fibers have dimension less than a specified integer. Jan 14 '18 at 21:40