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TitleGröbner bases for families of affine or projective schemes
Author(s) Michael Wibmer
TypeArticle in Journal
AbstractLet I be an ideal of the polynomial ring A[x] = A[x_1, ,x_n] over the commutative, Noetherian ring A . Geometrically, I defines a family of affine schemes, parameterized by Spec ( A ) : For p ∈ Spec(A), the fibre over p is the closed subscheme of the 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 〈σp(I)〉 . 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.
KeywordsGröbner cover, Canonical decomposition, Parametric polynomial system
ISSN0747-7171
URL http://www.sciencedirect.com/science/article/pii/S0747717107000624
LanguageEnglish
JournalJournal of Symbolic Computation
Volume42
Number8
Pages803 - 834
Year2007
Edition0
Translation No
Refereed No
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