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Nov 23 at 12:32 comment added Jason Starr Use the Noether Normalization Theorem to find a closed subvariety $Y$ of $\mathbb{A}\times X$ such that the restriction morphism from $Y$ to $X$ is dominant and generically finite. Now let $D$ be any divisor in $X$ that intersects the image of $\mathbb{A}^1\times z$ yet does not contain it. Let $E$ be the preimage of $D$ in $Y$, and let $Z'$ be the closure of the image of $E$ in $Z$.
Nov 22 at 15:24 history asked Biman Roy CC BY-SA 4.0