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 intersection.
Question 1: Wouldn't that imply that $\psi^{-1}(c)$ is also a complete intersection for any $c\not=0$?
Now consider a subscheme $Y\subseteq \mathbb{A}_{\mathbb{C}}^n$, where $Y$ is a complete intersection.
Question 2: If $Y\cap \psi^{-1}(0)$ is a complete intersection, then does that imply $Y\cap\psi^{-1}(c)$ is also a complete intersection, where $c\not=0$?
$ $ Here is the reason why I am thinking along the above lines. Consider $ \psi: X = \operatorname{Spec}\left( \dfrac{\mathbb{C}[x,y,z,w,a,b,c,d][t]}{ \left
Now take $ Y= \operatorname{Spec}\left( \dfrac{\mathbb{C}[x,y,z,w,a,b,c,d][t]}{ \left< xy+zw, x+a+c \right> } \right). $ Then isn't it true that $ Y\cap\psi^{-1}(0) = \operatorname{Spec} \left( \dfrac{\mathbb{C}[x,y,z,w,a,b,c,d][t]}{ \left< xy+zw, x+a+c, ab+cw+d^2 \right> } \right) $ while $ Y\cap\psi^{-1}(1) = \operatorname{Spec} \left( \dfrac{\mathbb{C}[x,y,z,w,a,b,c,d][t]}{ \left< xy+zw, x+a+c, ab+cw \right> } \right)? $ Please correct me if there is a typo anywhere in the example, or if some thought process isn't entirely correct.
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Any thoughts, counter-examples, or references would be great since I have limited deformation theory notes on hand.