Gröbner bases for families of affine or projective schemes

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 σp : 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. Introduction Let A be a commutative, noetherian ring with identity and A[x] = A[x1, . . . , xn] the polynomial ring in the variables x1, . . . , xn over A. We denote the residue field at p ∈ Spec(A) by k(p). Geometrically an ideal I ⊂ A[x] defines a family of affine schemes over Spec(A): The canonical map A → A[x]/I gives rise to a morphism of affine schemes φ : Spec(A[x]/I) → Spec(A). For p ∈ Spec(A) the fibre φ(p) is the closed subscheme ofAk(p) = Spec(k(p)[x]) determined by 〈σp(I)〉 where σp : A[x] → k(p)[x] denotes the trivial extension of the canonical map A → k(p). If I is a homogeneous ideal we analogously obtain a family of projective schemes from φ : Proj(A[x]/I) → Spec(A). The fibre φ(p) is the closed subscheme of Pk(p) = Proj(k(p)[x]), again determined by 〈σp(I)〉. For a chosen term order we wish to study – simultaneously for all p ∈ Spec(A) – the unique reduced Gröbner basis of 〈σp(I)〉. It is well known that such a Supported by the FWF (Project P16641)