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 $\mathcal{ O} = \{ 1,x,y,z,yz\}$.
Categories: math.AC