Graded Algebras over Polynomial Rings

Given a trivially graded polynomial ring $A=K[a_1,\dots,a_m]$ over a field $K$ and a positively graded polynomial ring $P=A[x_1,\dots,x_k]$, we study graded rings $R=P/I$, where $I$ is a homogeneous ideal in $P$ such that $I\cap A = \{0\}$. The corresponding morphism $\Theta: {\rm Spec}(R) \rightarrow {\rm Spec}(A) = \mathbb{A}^m_K$ is used to prove that ${\rm Spec}(R)$ is connected. Then we characterize and compute the following loci in $\mathbb{A}^m_K$: the set ${\rm Sing}_0(\Theta)$ of all points such that the corresponding point in the zero section of $\Theta$ is singular in ${\rm Spec}(R)$, the set ${\rm Sing}_v(\Theta)$ of all points $\Gamma$ such that the origin of the fiber $F_\Gamma$ of $\Theta$ is singular, and the set ${\rm Sing}_s(\Theta)$ of all points $\Gamma$ such that $\dim({\rm Sing}(F_\Gamma)) \ge 1$. These results are then used to study MaxDeg border basis schemes, as their coordinate rings are non-negatively graded by the total arrow degree and they have the required structure. In particular, we explicitly determine the singular loci for the $\mathcal{O}$-border basis schemes with $\mathcal{O}=\{1,x,y,z,z^2\}$ and $\mathcal{O} = \{1,x,y,z,yz\}$.