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Reducing a flat morphism $\psi:X\rightarrow\mathbb{A}_{\mathbb{C}}^1\;$ to $\;\psi|_{Y}: X\cap Y\rightarrow \mathbb{A}_{\mathbb{C}}^1$

Suppose $\psi: X\rightarrow \mathbb{A}_{\mathbb{C}}^1$ is a flat morphism, where $X\subseteq \mathbb{A}_{\mathbb{C}}^n$ with $X$ not needing to be smooth, with $\psi^{-1}(0)$ being a complete ...