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Since today is the Chow-variety day, I'm going to ask my question here.

Suppose I have a smooth projective variety $X$ over a field of characteristic zero, and a smooth hyperplane $p: H\hookrightarrow X$.

Is the pullback $p^* : C^m(X)\to C^m(H)$ between Chow varieties of $m$-codimensional cycles, a morphism of Chow varieties with property $\mathcal{P}$?

  • $\mathcal{P}$ = open immersion
  • $\mathcal{P}$ = étale
  • $\mathcal{P}$ = flat
  • $\mathcal{P}$ = closed immersion

I'm hoping for the third to be true. I'm not an expert on Chow varieties, but I need them for a technical lemma I'm after.

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    $\begingroup$ There is no morphism $p^*$. The intersection with the hyperplane section $H$ of a cycle of pure-dimension $n-m$ in $X$ need not have pure dimension $n-m-1$ in $H$. Presumably you want to restrict the morphism to the open subscheme parameterizing those cycles whose intersection with $H$ has pure dimension $n-m-1$. At any rate, typically $p^*$ satisfies none of the properties that you list. $\endgroup$ – Jason Starr Jan 15 '18 at 1:39
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Counterexamples. Here are examples showing that each of the properties above can fail. Let $X$ be a smooth cubic surface in $\mathbb{P}^3_k$, where $k$ is a field. Let $H$ be a smooth hyperplane section of $X$. This is a smooth, geometrically connected, projective curve of genus $1$ (a plane cubic).

The Fano scheme of lines on $X$, $F(X)$, is a finite, étale $k$-scheme of length $27$. The restriction morphism is everywhere defined on $F(X)$, $$p^*:F(X) \to H.$$ Since $F(X)$ is finite and $H$ is a curve, the image of $p^*$ contains no nonempty open subset. Thus, $p^*$ is not an open immersion, it is not étale, and it is not flat.

Finally, if $H$ contains any of the $135$ intersection points of a pair of lines of $X$, then $p^*$ is not injective on geometric points, thus it is not a closed immersion.

Sufficient Hypotheses for Flatness. You write that you want $p^*$ to be flat. There are several sufficient hypotheses for this to hold, although they are usually stated in terms of the Hilbert scheme or some other parameter space than the Chow variety. The point is that these hypotheses are usually stated in terms of infinitesimal deformation theory, and that is understood much better for the Hilbert scheme than the Chow variety (cf. the work of Angeniol and of David Rydh for the infinitesimal theory of the Chow variety).

Here is an example of a sufficient hypothesis that is useful for many applications: if $C\subset X$ is a closed subscheme that is a local complete intersection scheme of pure dimension $n-m$ whose intersection with $H$ has pure dimension $n-m-1$, if for every $q\geq 2$ the following cohomology group vanishes, $$H^q:= H^q(C,\textit{Hom}_{\mathcal{O}_C}(\mathcal{I}_{C/X}/\mathcal{I}_{C/X}^2,\mathcal{I}_{C\cap H/C})),$$ and if the fiber of $p^*$ at $[C]$ has dimension no greater than the "expected dimension", $$e=\chi(C,\textit{Hom}_{\mathcal{O}_C}(\mathcal{I}_{C/X}/\mathcal{I}_{C/X}^2,\mathcal{I}_{C\cap H})),$$ then $p^*$ is flat of pure relative dimension $e$ on an open subscheme of the Hilbert scheme that contains the point $[C]$ parameterizing $C$. Here $\mathcal{I}_{C/X}$ is the ideal sheaf of $C$ in $X$, and $\mathcal{I}_{C\cap H/C}$ is the ideal sheaf of $C\cap H$ in $C$.

Similarly, for $p^*$ to be smooth near $[C]$, replace the vanishing hypothesis for all $q\geq 2$ by vanishing for all $q\geq 1$. For $p^*$ to be étale, also add the hypotheses that the expected dimension $e$ equals $0$. If you also know that $p^*$ is generically injective (perhaps by a computation of the degree of $p^*$ in enumerative geometry), then $p^*$ is an open immersion. The morphism $p^*$ is unramified precisely if $H^0$ vanishes. If also $p^*$ is injective, then $p^*$ is a closed immersion.

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