2006/08/01 by Michael Wibmer, Wibmer, Michael
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.math/0608019
openalex publication_date 2006/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let I be an ideal of the polynomial ring A[x]=A[x1,...,xn] over the commutative, noetherian ring A. Geometrically I defines a family of affine schemes over \Spec(A): For \p∈\Spec(A), the fibre over \p is the closed subscheme of affine space over the residue field k(\p), which is determined by the extension of I under the canonical map σ_\p:A[x]→ k(\p)[x]. If I is homogeneous there is an analogous projective setting, but again the ideal defining the fibre is \sigI. For a chosen term order this ideal has a unique reduced Gröbner basis which is known to contain considerable geometric information about the fibre. We study the behavior of this basis for varying \p and prove the existence of a canonical decomposition of the base space \Spec(A) into finitely many locally closed subsets over which the reduced Gröbner bases of the fibres can be parametrized in a suitable way.