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I was reading Kollar's book "Rational curves on algebraic varieties". This is from Chapter IV, proposition $1.3$. I don't understand the proof from $1.3.3$ to $1.3.1$, Page- 182. Suppose I have a dominant morphism $\mathbb{A}^1 \times Z \to X$ over an algebraically closed field $k$, where $Z, X$ are smooth varieties. Also assume that for some $z \in Z$ the map $\mathbb{A}^1 \times z \to X$ is non-constant. Then how to prove there is some smooth variety $Z^{\prime}$ such that $\dim(Z^{\prime}) = \dim(X) - 1$ with a dominant morphism $\mathbb{A}^1 \times Z^{\prime} \to X$. I am trying to do using hyperplane section of $Z$. But how to show taking suitable hyperplane section of $Z$ the map remains dominant.
Any suggestion or hint is highly appreciated.

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  • $\begingroup$ 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 $Z$. $\endgroup$ Commented Nov 23 at 12:32

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