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Let $X$ be an integral affine scheme of finite type over a field $k$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get a $k$-scheme $X'$. Is it true that there is no closed immersion from a proper $k$-scheme of dimension $\geq n$ to $X'$?

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No, this is not true. Take $X = \mathbb{A}^4$ and let $$ Y = \{x_1x_3 - x_2^2 = x_1x_4 - x_2x_3 = x_2x_4 - x_3^2 = 0 \} \subset X $$ be the cone over a rational twisted cubic curve in $\mathbb{P}^3$. The codimension of $Y$ in $X$ is $n = 2$. On the other hand, it is easy to check that the fiber of the blowup $X' \to X$ over the origin (the vertex of the cone) is $\mathbb{P}^2$.

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  • $\begingroup$ what if we assume that $X$ and $Y$ are smooth? $\endgroup$ – rori Mar 8 at 12:25
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    $\begingroup$ Since $X$ is affine, any proper subvariety of $X'$ is contained in a fiber of the blow up, which has dimension $\leq n-1$ if $X$ and $Y$ are smooth. $\endgroup$ – abx Mar 8 at 13:19
  • $\begingroup$ @abx do you know a reference where this is described in some generality? I think there are examples with X singular and Y smooth where dimension inequality is satisfied, so there may be non-tautological necessary conditions for it to hold. $\endgroup$ – Aknazar Kazhymurat Mar 8 at 21:48
  • $\begingroup$ @Aknazar Kazhymurat: No, I don't know any reference, in fact I don't know any example of what you mention — but I am not an expert. $\endgroup$ – abx Mar 9 at 6:16

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