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Let $k$ be a field and $A$ a noetherian local $k$-algebra. Let $I$ and $J$ be two ideals of $A$ with $I^2 \subset J \subset I$. Let $A[tI] = \bigoplus\limits_{i\geq 0} t^i I^i$ and $A[tJ] = \bigoplus\limits_{i\geq 0} t^i J^i$ be the Rees algebras for $I$ and $J$, respectively.

Question: Is it true that $\mathrm{Proj}(A[tI]) \cong \mathrm{Proj}(A[tJ])$ as $k$-schemes? If not, then in what condition one can get such an isomorphism of $k$-schemes?

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    $\begingroup$ It seems like this is almost never true without some strong hypothesis; blowing up $(x,y)$ and $(x,y^2)$ in $k[x,y]$ give very different results, as the latter is singular and the former not. One case where it is immediately seen to be true is if I and J are both powers of another ideal. $\endgroup$ – Devlin Mallory Jun 25 at 12:43
  • $\begingroup$ Thanks you for the example. $\endgroup$ – user124771 Jun 26 at 13:47

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