Groebner bases for families of affine or projective schemes

الأصل · EN

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 (A): For ∈(A), the fibre over is the closed subscheme of affine space over the residue field k(), which is determined by the extension of I under the canonical map σ:A[x]→ k()[x]. If I is homogeneous there is an analogous projective setting, but again the ideal defining the fibre is. 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 and prove the existence of a canonical decomposition of the base space (A) into finitely many locally closed subsets over which the reduced Gröbner bases of the fibres can be parametrized in a suitable way.