Let $k$ be a field, $i:X\hookrightarrow \mathbf{P}(\mathscr{E})$ be a fixed projective embedding of a smooth projective $k$-variety $X$, whose dimension is pure and equals $d\ge 0$.

For $0\le p\le d$, denote by $C^p(X,i)$ the Chow $k$-variety of effective codimension $p$ algebraic cycles on $X$.

Fix affine codimension $n$ $k$-subvarieties $\mathscr{Y}:=\{Y_1,\ldots, Y_t\}$ for some fixed integers $n,t$, with $0\le n\le d$, and for an integer $0\le s\le d$ denote by $$W^p_{\mathscr{Y},s}(X,i)$$ the locus of $C^p(X,i)$ of those effective codimension $p$ $k$-cycles on $X$ each of whose components intersect each one the $Y_i$'s in $\mathscr{Y}$ in codimension at least $s$.

Is $W^p_{\mathscr{Y},s}(X,i)\subset C^p(X,i)$ an open subvariety? Is it closed?

  • 2
    $\begingroup$ Do you mean that the $Y_i$ are affine as schemes? In that case, the locus is neither open nor closed. Let $X$ be $\mathbb{P}^3$, let $Y_1$ be the complement of a line $L$ inside a (linear) $2$-plane $P$ in $X$, let $s$ equal $2$, and consider the component of $C^2(\mathbb{P}^3)$ parameterized curves of degree $2$. Your locus contains a dense Zariski open of $C^2(X)$. When the conic specializes to a union of two lines, one of the two lines might be contained in $P$. If that line equals $L$, you are in $C^2(X)$, otherwise not. $\endgroup$ Jan 14 '18 at 11:51
  • $\begingroup$ @JasonStarr Although it seems that slightly changing the question the OP asks: if $X$ is quasi-projective and of the form $X_1 \times V$, for $X_1$ smoothprojective and $V$ smooth affine, and if $Y$ is any closed subvariety of $X$ of the form $X_1\times W$, for $W$ an effective divisor on $V$, then for $\mathscr{Y} = \{Y\}$ and all the appropriate $p,s$, indeed the locus $W_{\mathscr{Y},s}^p(X,i)\subset C^p(X,i)$ is open. Here by $C^p(X,i)$ I mean the Chow variety of the quasi-projective variety $X$ under $I$, itself quasi-projective. I came across this question myself. Am I missing something? $\endgroup$
    – user113452
    Jan 14 '18 at 12:35
  • $\begingroup$ I'd follow up to the OP's question and ask: are there methods to check openness of loci in Chow varieties? (eg. For $p=1$ I guess one can try to rephrase the pb into a problem about openness of an appropriate locus in the Picard scheme, to be then addressed by coherent cohomology methods.) $\endgroup$
    – user113452
    Jan 14 '18 at 12:39

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