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Let $X$ sit in some affine space. Suppose $f: X\rightarrow \mathbb{C}$ is a flat family and the fiber over some nonzero point is a complete intersection. Then isn't it always true that $f^{-1}(0)$ is also a (globally) complete intersection?

That post discusses that the fibers that which are complete intersections may not extend to all of $T$ (they take the base space to be $T$, rather than $\mathbb{C}$).

But I thought the whole idea of flatness is that the dimension of each fiber doesn't jump up. I'm guessing that over some points in $T$ or $\mathbb{C}$, the dimension can go down?

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Note: my base space is integral and $1$-dimensional while the base space $T$ in that post may not be. Would that be the reason why locally complete intersection property in his case is an open restriction while globally complete intersection is a closed condition?

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