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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 } \right) \longrightarrow \operatorname{Spec}\mathbb{C}[t]. $ Then not only is $\psi^{-1}(0)$ a complete intersection, $\psi^{-1}(1)$ is also a complete intersection.

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.

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    I will repost my notion of complete intersection and in which space soon. What I wanted to do in this problem is come up with a flat map for this question http://math.stackexchange.com/questions/166341/explicitly-constructing-a-certain-flat-family#comment383859_166341 using a slightly different strategy so that we obtain $\phi^{-1}((0,0,0))$ and $\phi^{-1}(t_1, t_2, t_3)$ (for some fixed $(t_1, t_2, t_3)$) as in that problem.2012-07-04

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You should make assumptions about the relative dimension of the new intersection over the target.

Anyway, you should also remember the following. (See Hartshorne, Algebraic Geometry) Whenever we have a morphism of schemes, say, of finite type over a field

$f:X\to C$ with $C$ a smooth curve and $X$ equidimensional, the morphism is flat if and only if no irreducible component of $X$ is contained within a fibre.

Milne, Etale Cohomology, in Chapter I, has a section on flat morphisms where this result is stated and, if I'm not mistaken, proven (or a reference is given). Another beautiful reference is Mumford's The Red Book of Varieties and Schemes.