  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 over $\Spec(A)$: For $\p\in\Spec(A)$, 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 $\sigma_\p:A[x]\to k(\p)[x]$.
If $I$ is homogeneous there is an analogous projective setting, but again the
ideal defining the fibre is $\sigI$. For a chosen term order this ideal has a
unique reduced Gr\"{o}bner 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 Gr\"{o}bner bases of the fibres can be parametrized in a suitable way.
