Graded Algebras over Polynomial Rings

Given a trivially graded polynomial ring $P_0=K[a_1,\dots,a_m]$ over a field $K$ and a positively graded polynomial ring $P=P_0[x_1,\dots,x_s]$, we study graded rings $R=P/I$, where $I$ is a homogeneous ideal of $P$ satisfying $I\cap P_0=\{0\}$. The induced morphism $Θ: {\rm Spec}(R)\rightarrow {\rm