Graded Algebras over Polynomial Rings
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915818813521920 |
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| author | Kreuzer, Martin Robbiano, Lorenzo |
| author_facet | Kreuzer, Martin Robbiano, Lorenzo |
| contents | 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 $Θ: {\rm Spec}(R) \rightarrow {\rm Spec}(A) = \mathbb{A}^m_K$ 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 ${\rm Sing}_0(Θ)$ of all points such that the corresponding point in the zero section of $Θ$ is singular in ${\rm Spec}(R)$, the set ${\rm Sing}_v(Θ)$ of all points $Γ$ such that the origin of the fiber $F_Γ$ of $Θ$ is singular, and the set ${\rm Sing}_s(Θ)$ of all points $Γ$ such that $\dim({\rm Sing}(F_Γ)) \ge 1$. 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 $\mathcal{O} = \{1,x,y,z,yz\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22891 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Graded Algebras over Polynomial Rings Kreuzer, Martin Robbiano, Lorenzo Commutative Algebra 13A02 (Primary) 14E25, 14Q20, 13P10 (Secondary) 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 $Θ: {\rm Spec}(R) \rightarrow {\rm Spec}(A) = \mathbb{A}^m_K$ 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 ${\rm Sing}_0(Θ)$ of all points such that the corresponding point in the zero section of $Θ$ is singular in ${\rm Spec}(R)$, the set ${\rm Sing}_v(Θ)$ of all points $Γ$ such that the origin of the fiber $F_Γ$ of $Θ$ is singular, and the set ${\rm Sing}_s(Θ)$ of all points $Γ$ such that $\dim({\rm Sing}(F_Γ)) \ge 1$. 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 $\mathcal{O} = \{1,x,y,z,yz\}$. |
| title | Graded Algebras over Polynomial Rings |
| topic | Commutative Algebra 13A02 (Primary) 14E25, 14Q20, 13P10 (Secondary) |
| url | https://arxiv.org/abs/2602.22891 |