# Relation between $\mathrm{Proj}(A[tI])$ and $\mathrm{Proj}(A[tJ])$ for ideals $I$ and $J$ of $A$ with $I^2 \subset J \subset I$

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?

• 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. – Devlin Mallory Jun 25 at 12:43
• Thanks you for the example. – user124771 Jun 26 at 13:47