Let $X$ be an integral affine scheme of finite type over $\mathbb{C}$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get a scheme $Bl_Y X$. Let $m_X(Y)$ be the maximum dimension of a closed subscheme of $Bl_YX$ that is proper (as a $\mathbb{C}$-scheme). If we fix $n$, and let $X$ and $Y$ vary, is it true that $m_X(Y)$ has a uniform upper bound?
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1$\begingroup$ I mean, intuitively the only proper subschemes of this gadget will be the fibers of the exceptional divisor (and their subschemes), which in the case of $Y$ smooth are projective spaces of dimension $n - 1$. But perhaps things get weird when $Y$ is singular? $\endgroup$– Tabes BridgesMar 30, 2019 at 20:24
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$\begingroup$ @TabesBridges thank you for your suggestion. $Y$ is not necessarily smooth. $\endgroup$– user74900Mar 30, 2019 at 20:26
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1$\begingroup$ Have you calculated what happens when, e.g. you blow up $\mathbb A^3$ along a pair of intersecting lines? Because it seems to me the question is, can anything strange happen with the structure of the exceptional divisor? Since it is, in fact, a divisor, I would assume no. $\endgroup$– Tabes BridgesMar 30, 2019 at 20:29
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