Let $X$ be an irreducible smooth projective variety of pure dimension $d$ over the complex numbers and $Z\subset X\times X$ a codimension $d$ irreducible smooth closed subvariety.



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> Is there a smooth hyperplane section $H$ on $X$ such that $H\times X$ contains a cycle in the rational equivalence class of $Z$?
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I expect the answer to be "no". Is it "yes" if one drops smoothness of $H$?



Cycles contained in ample enough hypersurfaces
Let $X:=\Sigma\_g$ be a smooth curve of genus $g\geq 2$ and $Z=\Delta$, where $\Delta \subset \Sigma\_g \times \Sigma\_g$ is the diagonal.


Then $\Delta$ is not rationally equivalent to a union of fibres of $p\_1 \colon \Sigma\_g \times \Sigma\_g \to \Sigma\_g$, for instance because $\Delta^2=2-2g <0$.


