.  -   Since $V$ is unmixed, its top dimensional part $C$ is equal to $V$. Hence $Z \subset C$.  -   $\langle (I_{Z})_{\leq d} \rangle = I_C$ and $I_C$ is [$d$-regular]{}.  -   [**If we also assume that**]{} $h^0({\mathcal I}_{C_{red}}(d-0)) = 0$ then $V$ is reduced and $C = C_{red}$ is $d$-regular.  W