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Suppose that I have a variety $X$, and a set of subvarieties $A_1,...,A_r$ of codimension $n$ and a set of subvarieties $B_1,...,B_s$ of codimension $m$. Is there a nice way to determine whether there's a subvariety $Y$ such that $A_i \cap Y$ is of codimension $<n$ in $Y$ and $B_i \cap Y$ is of codimension $m$ in $Y$?

In the particular case I am thinking of, $X$ is simply affine space of dimension $n$, and the $A_i$ and $B_i$ are all subspaces containing the origin.

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    $\begingroup$ What kind of answer are you looking for? There are obvious counterexamples, e.g., $r=s=1$, $m=n$ and $A_1$ equals $B_1$. Even in situations where you might expect a positive answer, singularities of $X$ can cause problems, e.g., $X$ is an affine cone over a smooth quadric surface $S$, and $A_1$ and $B_1$ are cones over linearly equivalent lines in $S$. You need to specify your situation. $\endgroup$ Jan 12, 2016 at 12:11

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