.  Let $C$ be the unmixed one-dimensional part of $V$. Let $Z_0$ be the subscheme of $Z$ lying on $C$ (defined by $[I_Z + I_C]^{sat}$) and $Z_0$ the residual scheme (defined by $[I_Z : I_C]$). Then  -   $\langle (I_{Z_0})_{\leq d} \rangle = I_C$ and $