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, parameterized by : For , the fibre over is the closed subscheme of the affine space over the residue field , which is determined by the extension of I under the canonical map . 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 into finitely many, locally closed subsets over which the reduced Gröbner bases of the fibres can be parametrized in a suitable way.