Graded Algebras over Polynomial Rings
Martin Kreuzer, Lorenzo Robbiano

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 over a field and a positively graded polynomial ring , we study graded rings , where is a homogeneous ideal in such that . The corresponding morphism is used to prove that is connected. Then we characterize and compute the following loci in : the set of all points such that the corresponding point in the zero section of is singular in , the set of all points such that the origin of the fiber of is singular, and the set of all points such that . These results are then used to study MaxDeg border basis schemes, as…
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Taxonomy
TopicsPolynomial and algebraic computation · Commutative Algebra and Its Applications · Algebraic Geometry and Number Theory
