Groebner bases for families of affine or projective schemes

Abstract

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.