Graded Algebras over Polynomial Rings

TL;DR

This paper investigates the structure and singularities of graded rings over polynomial rings, providing methods to compute singular loci and applying these to border basis schemes with explicit examples.

Contribution

It introduces a framework for analyzing graded rings over polynomial rings, characterizes their singular loci, and applies these results to border basis schemes with explicit computations.

Findings

Connectedness of ${\rm Spec}(R)$ established.

Explicit singular loci for specific border basis schemes computed.

Framework applicable to analyzing singularities in graded algebraic structures.

Abstract

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 $\mathcal{O} = \{1,x,y,z,yz\}$.