Resolutions of ideals of fat points with support in a hyperplane

Let $Z'$ be a fat point subscheme of $\mathbb{P}^{d}$, and let $x_0$ be a linear form such that some power of $x_0$ vanishes on $Z'$ (i.e., the support of $Z'$ lies in the hyperplane $H$ defined by $x_0=0$, regarded as $\mathbb{P}^{d-1}$). Let $Z(i)=H\cap Z'(i)$, where $Z'(i)$ is the subscheme of $\mathbb{P}^{d}$ residual to $x_0^i$; note that $Z(i)$ is a fat points subscheme of $\mathbb{P}^{d-1}=H$. In this paper we give a graded free resolution of the ideal $I(Z')$ over $R'=K[{\mathbb{P}}^{d}]$, in terms of the graded minimal free resolutions of the ideals $I(Z(i))\subset R=K[{\mathbb{P}}^{d-1}]$. We also give a criterion for when the resolution is minimal, and we show that this criterion always holds if $\hbox{char}(K)=0$.