Grobner bases for families of affine or projective schemes

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 is an element of 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 sigma p : A[x] --> k(p)[x]. If I is homogeneous, there is an analogous projective setting, but again the ideal defining the fibre is <sigma(p)(I)> For a chosen term order, this ideal has a unique reduced Grobner 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 Grobner bases of the fibres can be parametrized in a suitable way. (C) 2007 Elsevier Ltd. All rights reserved.