Details:
Title  Gröbner bases for families of affine or projective schemes  Author(s)  Michael Wibmer  Type  Article in Journal  Abstract  Let 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.  Keywords  Gröbner cover, Canonical decomposition, Parametric polynomial system  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717107000624 
Language  English  Journal  Journal of Symbolic Computation  Volume  42  Number  8  Pages  803  834  Year  2007  Edition  0  Translation 
No  Refereed 
No 
