Groebner bases for families of affine or projective schemes

摘要

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 < σ_p(I) >. For a chosen term order, this ideal has a unique reduced Groebner 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 Groebner bases of the fibres can be parametrized in a suitable way.