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Let $X$ be a flat family of varieties over a smooth projective variety $V$. Consider a subsheaf $I\subset\mathcal{O}_X$.

1) what is the definition of a flat family of ideal sheaves?

2) Is this a flat family of ideal sheaves on $Y$?

3) Also a flat family of ideal sheaves on $Y$ gives an ideal sheaf of the closed subvariety over each fiber. Do these closed subvarieties have to be equidimensional?

For example consider a line bundle $A$ on $X$ and independent global sections $a_1,...,a_r$. This gives a morphism from $L^{-r}\rightarrow O_X$ whose image is the ideal of the closed subscheme defined by the sections on $X$. This gives a closed subschemes over each fiber. These closed subschemes may not be of the same dimension. But Will this image sheaf be a flat family of ideal sheaves?

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